The Chelsea left-back took to the social networking website on Friday after the governing body called his reliability into question when supplying evidence on behalf of John Terry in the recent independent regulatory commission's hearing into the alleged racial abuse towards Anton Ferdinand.
Cole wrote on his Twitter account: "Hahahahaa, well done #fa I lied did I, #BUNCHOFT****."
Cole, who later deleted the tweet and apologised "unreservedly" in an official statement, has been included in the Three Lions squad to face San Marino and Poland in the upcoming World Cup qualifiers and remains a part of Roy Hodgson's group for the two fixtures.
And Bernstein, talking on BBC Radio 5 Live, confirmed that Cole has personally apologised to him on Monday night and is now "free to play for England" again, with only Roy Hodgson able to decide against his selection.
He said: "He apologised immediately on Friday and he came to see me last night and apologised to me personally," said Bernstein.
"He showed real contrition. He said he was really sorry.
"He is free to play for England over the coming matches. It is up to the manager to decide whether he plays or not."