The England boss is satisfied the problem has been sorted after the experienced left-back apologised for his outburst criticising the FA and is now ready to move forward
Chelsea left-back Cole took to Twitter last week to criticize the FA's findings in their hearing into John Terry’s incident with QPR’s Anton Ferdinand last year, posting: "Hahahahaa, well done #fa I lied did I, #BUNCHOFT****."
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He swiftly apologised for the post, but speculation had suggested Cole could be dropped from the squad to face San Marino at Wembley.
However, Bernstein revealed Cole had "showed real contrition" when apologising for his action, 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 realises 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 double-header.