Back in 1916, the French wanted to equip their troops with an automatic rifle, among other projects they had the Ribeyrolles 1918 Automatic Carbine, which didn't use the then standard 8mm Lebel, but a necked down Winchester .351 which would be lighter to carry and have significantly less recoil than a full-blown rifle cartridge.
Ditto for Vladimir Fyodorov's Fedorov Avatomat, a 1915 design that didn't use a proprietary cartridge, but the 6.5mm Arisaka, which was less powerful than other rifle cartridges.
Right before WWII, the US military was looking into equipping their troops with a lightweight weapon that would be more effective than the 1911 pistol and lighter than either the M1 Garand or the M1 Thompson. They came up with the M1 Carbine, which used its own .30 ammo.
During WWII the Germans introduced a compact, reduced load 7.92mm Kurz bullet, designed a new rifle with automatic fire capability which is generally considered the ancestor of the assault rifle.
The Soviets came up with their own variant, the AK-47, which used a 7.62x39 bullet which was less powerful than the standard 7.62x54 rifle bullet and therefore easier to control on semi-automatic fire.
The British, understanding the limitations of their own .303 ammo decided they would replace it with a new .280 intermediary cartridge. FN in Belgium had similar plans and was working on the FN FAL.
Pretty much everyone had figured out that if you wanted to give soldiers an automatic weapon, you needed a weapon that could fire an intermediary cartridge so that the useful feature called "automatic fire" would be viable.
And in a haze of a testosterone-filled fever dream, fuelled by steaks, Cuban Cigars and an elephant hunt, the US came up with the full-blooded 7.62x51mm, because Uncle Sam would rather be damned than to give his brave boys anything less than a turbocharged bullet that could blow a commie in half a mile away !
It didn't matter that the M14 rifle was a serious threat to the sky after two shots, it was the whole hog or nothing and the rest of NATO better comply or else …
The British dropped the .280 and FN reconfigured the FAL to 7.62mm, despite having designed it for a much lighter cartridge, the weapon went on to become the right arm of the free world.
The M14 was introduced in 1959, three years previously, Eugene Stoner and a team of clever engineers came up with a lightweight rifle that used plastics and aluminium parts and most importantly, used an intermediary cartridge.
In 1960 Curtis LeMay was sufficiently impressed by this new rifle that he ordered some for the Air Force.
A few years later the US army finally gave up on giving their infantry portable autocannons and switched to the M16 rifle in 5.56mm.
Now, until then most intermediate cartridges hovered around the .30 mark, 6.5 to 7.5mm. Several 6.5mm calibers were found to have a very high ballistic coefficient. The 5.56mm was relatively underpowered and fired a small, lightweight bullet.
There were some concerns that such a light bullet might not be very effective, but the marketing department came up with a brilliant sales pitch. They filled plastic and metal containers with water, sealed them tight and shot them with 5.56mm bullets. If a container filled to the brim with incompressible water explodes when hit by a high velocity bullet, imagine what will happen when a human, which is 70% water, when hit.
In short the 5.56mm rifle was sold in the idea that if you hit a commie in the pinkie finger, his kidneys would explode because of some pretty funky voodoo magic called "Hydrostatic shock"
Meanwhile the Soviets, who didn't have a marketing department nor a bunch of brass that needed to be sold a fantasy, decided that they would also drop their intermediate caliber a notch and went for the seemingly even smaller 5.45mm bullet.
Note there is a world of difference between 5.56mm and 5.45mm. A simple side-by-side comparison quickly tells you that the Soviet design uses a much longer bullet, longer bullet means physically larger, means more mass. The longer longitudinal axis means it's more stable in flight, but while it has more mass it is inherently unstable because it's still a very light bullet by any standards going at very high speed. When it hits anything it's like an F1 car that hit a pebble and has gone out of control, it flies off in a random direction. The result is a bullet that performs like an Olympic gymnast cartwheeling inside your body and tears it up due to the long bullet.
The 5.56mm behaves much in the same way, except that it's shorter, lighter and does the most damage when it fragments.
But why switch to the 5.56mm in the first place you ask ? Smaller, lighter bullets means a lighter rifle, you can carry more bullets and a soldier with a lighter rifle and more bullets is a happy soldier.
Solid logic, except for one tiny problem called long-range performance. You see back in WWII they figured that the majority of firefights occurred within 300m, which means that the old ammo from the age of the bolt action rifle that could nail an enemy soldier a mile away was a waste of time and energy as most soldiers couldn't even hit a target that far.
The undue bout of nostalgic madness that overcame the US military in the 1950s and prompted them to ignore what everyone else on the planet had figured out was only compounded by the overcompensation in picking the rather weak (but still deadly) 5.56mm which doesn't perform very well beyond 300m.
So NATO is stuck between the overpowered and obsolete 7.62mm and the underpowered 5.56mm which has major issues like barrier penetration and long range effectiveness.
In the 2000's some clever designers came up with an answer, they used a case the same size as a 5.56mm and mated it with a long 6.x mm bullet with superior ballistics, rivaling and even exceeding the 7.62mm in performance.
Every gun expert figured that the 6.x would soon replace the 5.56mm in the US arsenal as this was the most logical choice, by retaining the same cartridge dimensions any 5.56mm rifle could be converted by simply swapping out the barrel and maybe a few minor modifications to the receiver and chamber. A new, better bullet, which didn't require introducing a whole new rifle, how could this possibly fail ?
By now you'll have realized that the US military's ability to jam a crowbar in even the best of plans is beyond compare. The first M16 hadn't even reached the hands of the first soldier when they were already planning a replacement. The list is nearly endless, it includes the SPIW, SALVO, the ACR, the OICW, the Individual Carbine project etc. Each and every one of these projects promised to revolutionize warfare and get the average rifleman to a completely new level. Each and every one died a slow death hemorrhaging money.
And now we finally come to the OP, there have been a zillion opportunities to come up with something better than the current service rifles. Logically you'd simply update existing M16 (or H&K 416 and FN SCAR) and M4 weapons with a conversion kit to 6.x mm and the problem would be solved. But it wouldn't be the US army if they didn't spend a fortune to throw money down a well and still get nowhere.
Odds are they fiddle around and come up with some Frankenstein creation in 5.56mm, based on the H&K 416, most likely a featherweight lower and receiver M4-style with a folding stock mated to a longer heavy barrel for improved long range fire while still keeping it compact enough to clear a phone booth or a really small closet.
They will not go for a bullpup or anything that varies from the M16/AR-15 platform because far too many soldiers have grown up shooting rifles that have the same controls as military rifles, they will continue to try to stretch the 5.56mm as far as it goes because switching to a new caliber is not an option.