"Having had time to consider my actions, I acknowledge now that the timing and manner of what I said was ill-judged," he said in a statement released by his representatives, Base Soccer Agency.
"My words were not meant to offend and for that I would like to apologise to the chairman, manager, my teammates and the fans," said the 27-year-old. "I'd also like to wish the lads good luck ahead of Sunday's game against Newcastle."
Rose, who has been sidelined with a knee injury for six months, sparked a debate after the unusually forthright interview in The Sun newspaper in which he said he would discuss moving to another club if an offer came in.
"As with everyone else in my team, in my opinion, I am worth more than I am getting," he told The Sun.
"I'm not saying I want out but if something came to me that was concrete, I'd have no qualms about voicing my opinions to anyone at the club.
"One thing is for sure — I know my worth and I will make sure I get what I am worth.
"I am not playing as well as I have done not to get paid what I think I am worth.
"I am not speaking on behalf of other players, but that is my view."
The Evening Standard newspaper reported his wages to be 70 000 pounds per week at a club that operates within a tight salary structure. Multiple media reports suggest Rose could double his wages by moving elsewhere.
Earlier this summer, his fellow fullback Kyle Walker joined Manchester City in a move which Rose, who is under contract until 2021, said was "a big blow."
Rose added that he hoped the club, who have not signed anyone in the transfer window, will bring in new players and "not ones you have to Google and say, 'who's that?'
Rose's manager Mauricio Pochettino later said the player had apologised and "it is time to move on."
"Of course he has a future and I want to keep him here. Nothing has changed on my perception of him," Pochettino told a press conference on Friday.
Rose's remarks have sparked a debate in the game in which many clubs insist their players do not stray into controversy.
"I'm surprised at Danny," said his former manager Harry Redknapp. "It's up to the manager and the owners whether they spend money really, and buy players," he wrote in his newspaper colum