The Bees have offered an initial £16m plus £4m in potential add-ons as they look to strengthen their attacking options for a second season in the Premier League.
Brentford failed with bids for Lewis-Potter last summer and in January and are now making a renewed push for the 21-year-old, who scored 12 goals in the Championship last season.
But Hull want Lewis-Potter to stay and lead a promotion bid next season.
“There is a firm offer but we're not doing anything yet,” Hull vice-chairman Tan Kesler told BBC Humberside Sport. “It's not enough. He's such a good guy and we did what we had to do for him.
“With Keano, the most important thing is that he is from here, and he has one more final step to achieve in Hull. If he believes in that, no problems. If he doesn't believe in that, then no problems and if the offer is right we will consider it.
“I'm not saying we're considering selling him. The market is open, we'd like to hear, we'd like to talk and that doesn't mean we're going to decide.
“I'd really like him to stay one more year and take us to the play-offs, if not the promotion dream that we have."
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