Leicester are bottom of the Premier League and suffered a sixth straight loss at Tottenham on Saturday, with Rodgers position under threat following a dismal start to the season.
The 6-2 loss at Spurs made it 22 goals conceded already this season, a record at this stage of a Premier League campaign.
According to the Daily Mail, Frank is the Foxes’ first choice to replace Rodgers and the club are pricing up the cost of sacking the Northern Irishman and bringing in the Brentford boss.
Rodgers’ lucrative contract means it would cost Leicester around £10m to dismiss the 49-year-old and the club are cautious given the need to meet Financial Fair Play rulings.
The under-pressure boss spoke on his future at the weekend and admitted he would respect any decision the board made.
“I don’t know, to be honest,” Rodgers told Sky Sports on his future. “Whatever their [the owners’] decision is, I will always respect it.
“I think we’ve seen enough. I felt the team tonight, if you watched them play in the first half, you could see the confidence and creativity. It’s just cutting out mistakes.
“There is no doubt we’ve had a tough first seven games, especially after the summer we’ve had – Tottenham away, Arsenal away, Chelsea away, Manchester United at home and a couple of other games we should win.
“It [the international break] has probably come at a good time to reset everything, break the cycle from the summer as well. The owners will do what it is they feel they need to do.
“I’m not daft, I know football, and the last six games don’t make great reading. But I have every confidence the team can push on and climb the table if they play like they did in large parts and cut out mistakes.”