Cole made the approach on Monday.
Hodgson told a press conference at the official opening of the National Football Centre this afternoon, televised by Sky Sports News< /i>: "I'm pleased the chairman has gracefully accepted the apology from Ashley. I know how saddened he was by it and I'm glad it's been resolved."
Hodgson added: "He's played 98 times (for England) and hardly missed a game when he's fit.
"He has been a regular for many years as one of the best, if not the best, left-back in the world. No-one I don't think who is fair-minded could criticise his commitment to the England shirt.
"His contrition is greater because he realises that he might have put his position in jeopardy - and I'm very happy he hasn't done that.
"The important thing for me was that the matter got resolved. He immediately tried to rectify the mistake and apologised for it.
"I always hoped this would be sorted out and I'm very happy with the outcome."