The former England international is one of several high-profile members of the Tigers’ squad into the final few weeks of their contracts.
Forest remain keen on taking Dawson back to the City Ground after failing with a deadline day approach last January.
At one stage, the deal looked set to go through only for Hull to pull the plug at the eleventh hour when unable to line up a replacement.
“I want Michael to be here, Michael knows that,” said Adkins. “All I can do is make Michael aware that I want him at the football club. All Michael can do is work very hard every day in training, set the right example as captain and he does that.
“The business side of things gets taken care of and, like anyone, he’ll look to get the best deal for himself. That’s what you should do.
“That might just be monetary, it might be length of contract and all sorts of things for each player.
“You have to put yourself in a player’s position. They’ve got to do right for them as individuals and their family so long as in the meantime they’re giving them all.
“He has been exemplary in that respect. I can’t speak highly enough of him.”
Dawson missed the two games after his proposed deal to Forest had collapsed through injury, the second being City’s 2-0 win at the City Ground.
Since then, however, he has been a mainstay of the team that has plotted a path to within touching distance of survival.
Last weekend’s 1-0 defeat at home to Sheffield Wednesday was only the second time Dawson has been absent in the last 13 games, hence why Adkins wants him to stay.
Aitor Karanka, however, is understood to be keen to revive the deal and bring in a player who has made 130 appearances for Hull in four years at the club.
“I can’t affect the messages coming out of their club (Forest), I can only talk about us,” he added. “Michael is our captain and he conducts himself in a spot-on manner.
“Every single day on the training ground, he gives 100 per cent to the team. He is in the latter stages of his career and he has been very successful here.
“Everyone should remember that. He has got to do the right business for him and his family.”
City, meanwhile, head to Nottingham knowing victory will guarantee the club’s Championship status, even if third bottom Barnsley also win at Leeds United.
Nine points separate the two Yorkshire clubs with both having four games remaining but Hull’s goal difference is vastly superior to that of the Reds.