Jesse James says most of the claims alleged in recent reports that he cheated on wife Sandra Bullock are false.

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"The vast majority of the allegations reported are untrue and unfounded," said James in a Thursday statement obtained by People.

"Beyond that, I will not dignify these private matters with any further public comment."

The Oscar-winning actress left the house she shares with the Jesse James is a Dead Man and former Monster Garage star on Monday -- two days before reports that he cheated on her surfaced.

James, who also competed on The Celebrity Apprentice's 2009 edition, allegedly carried out the 11-month affair with tattoo model Michelle "Bombshell" McGee while Bullock was in Atlanta filming The Blind Side -- for which recently won the Oscar for Best Actress.

In addition, McGee has stated James told her he was separated from Bullock.

In his statement, James also apologized to Bullock -- whom he wed in 2005 -- and his three children from his two previous marriages.

"There is only one person to blame for this whole situation, and that is me. It's because of my poor judgment that I deserve everything bad that is coming my way," he said.

"This has caused my wife and kids pain and embarrassment beyond comprehension and I am extremely saddened to have brought this on them. I am truly very sorry for the grief I have caused them. I hope one day they can find it in their hearts to forgive me."