Arsene Wenger reportedly had a bid of £30m plus adds-on rejected by the Liverpool board and Rodgers is determined to hold onto one of his prize assets.
Skipper Steven Gerrard signed a two-year contract extension on Monday and Rodgers inists Suarez will once again be lining up alongside him next season, "unless something drastic happens".
"He is a wonderful player and he is still very much a Liverpool player. No matter who the player is, nobody is bigger than Liverpool Football Club," said Rodgers.
"'Luis is a very important member of this squad and I am looking forward to working with him again this season.
"There has been lots of speculation about Luis moving to another club but, as I said, he is very much a Liverpool player.
"We had an offer that was nowhere near what we value him at. Of course he wants to work and play at the top level.
"But unless something drastic happens, he will be staying here. He has got three years left on his contract."
As for the season ahead, Rodgers is confident they can mount a serious challenge for a top-four spot in the Premier League in 2013/14.
"The Champions League is a competition we're fighting to get in," he added.
"The great name of Liverpool as a club and the history of the club dictate that we should be in that competition. There's no other measure for us.
"The challenge is very difficult because of the competitors now in England. We have to continue to fight.
"It was very difficult for us last season with a new infrastructure and a new way of working and playing, but I believe this season we can put up a great fight.
"Let's hope we can get back to where the club belongs."
Liverpool play an Indonesian All-Star XI on Saturday, although Suarez, along with defender Sebastian Coates and goalkeeper Pepe Reina, will not be involved as they enjoy an extended break following their involvement in the Confederations Cup.
Meanwhile, 20-year-old Reds defender Andre Wisdom, who is also part of the tour squad, has indicated that he may be willing to go out on loan.
"The manager has told me to consider it and keep it in mind," Wisdom told the club website.
"With us not being in the Europa League, the opportunity of playing every week is going to slim down a bit.
"Going on loan is an option. I'm open to any idea that is positive."