Did Soap and 141 have any information about who Prisoner 627 was?
Solution 1:
The prisoner was only known to them as "Makarov's nemesis", which they might have reasoned was Price based on Makarov being Zakhaev's right-hand-man (who had an arm blown off by Price in the famous Pripyat level of the original MW).
It seems unlikely however that they knew or really suspected who prisoner 627 was. Although Captain MacTavish brought along Price's 1911 pistol, he pointed it straight at Price's head before realizing the prisoner's identity.