The England boss is satisfied the problem has been sorted after the experienced left back apologized for his outburst at the FA.
Cole took to Twitter last week to criticise the FA's findings following their hearing into John Terry’s incident with QPR’s Anton Ferdinand last year, posting: "Hahahahaa, well done #fa I lied did I, #BUNCHOFT****."
He swiftly apologized for the post, but speculation had suggested Cole could be dropped from the squad to face San Marino at Wembley Stadium. Bernstein, though, revealed Cole had "showed real contrition" when apologizing for his actions, and Hodgson is satisfied the 98-capped defender is still fully committed to playing for his country.
"The chairman has gratefully accepted the apology," Hodgson told reporters. "I am quite relieved as England manager that this has now been sorted out between the chairman and Cole. "No one who is fair-minded could ever question his commitment to England and the England shirt. His contrition is the greater, because he realizes he may have put his position in jeopardy, and I'm very grateful he hasn't done that."
The Three Lions host San Marino on Friday evening before travelling to face Poland in a 2014 World Cup qualification doubleheader.