Cook's position came under further scrutiny on Monday following the team's 95-run defeat to India in the second Test at Lord's.
However, the opening batsman insisted after the match that he would not be stepping down from his role voluntarily and that he is eager to turn England's fortunes around.
"He's standing tall and refusing to buckle. It's brilliant from him," Swann, who retired at the end of 2013, told BBC Sport.
"He has an inner steel, stronger than anyone else I've ever known. He's going to fight like hellfire to change it round."
The defeat to India means that England have failed to win in their last 10 matches.