While Liverpool have not made a bid for Dempsey, Rodgers said in a press conference last month that "Clint is a player we've inquired about, it is as simple as that.
"Ian Ayre, our managing director, has spoken with the club to see what the position is," Rodgers went on. "That is where we're at. He's a very talented player, but we don't like to talk about other clubs' players."
The complaint is believed to have been made following those comments.
"We have received a complaint from Fulham regarding comments made by Liverpool FC officials and are looking into the matter," a Premier League statement read.
Dempsey was left out of Fulham's squad for their opening Premier League match of the last weekend as manager Martin Jol claimed the 29-year-old was no longer committed to the club. It has already been confirmed that Dempsey won't be selected for Fulham's next match against Manchester United on Saturday.
There is plenty of other speculation involving Liverpool as the close of the transfer window approaches. Marca has reported that Liverpool are now poised to sign Real Madrid midfielder Nuri Sahin on a season-long loan.
The Turkey international had been set to join Arsenal, but a disagreement has put the Reds in the driving seat. The move to Arsenal is said to have fallen through with the Gunners asking for a permanent move, something neither Sahin nor Real desired.
According to Marca, Sahin's agent, Reza Fazeli, met with Liverpool's representatives last night to finalize the terms of a deal.
Despite already bringing in Joe Allen this summer, Liverpool have also been linked to another midfielder in Hector Herrera. The 22-year-old has enjoyed a stunning summer having been voted the best player of the competition as Mexico won the prestigious Toulon youth tournament for the first time before helping his country to win Olympic gold at the London Games.
His Mexican club, Pachuca, have admitted that they have received approaches from both Liverpool and rivals Manchester United.
"There is an official interest. We have the document, but there's nothing else to add at this time," said sporting director Andres Fassi, according to the Daily Telegraph.
"We want him to continue in Pachuca," he added. "We know he's going to emigrate to England, though. Manchester United and Liverpool have talked to us.
"At whatever moment he wants, he can leave Pachuca and the club will give him the opportunity to do so."
Finally, Brendan Rodgers has dismissed the notion that striker Andy Carroll could be allowed to leave on loan before the end of the transfer window. The England international has been linked with a move away from Anfield since Rodgers hinted that Carroll would not figure in his plans this season.
Reports had suggested that Carroll's former club Newcastle were eager to take the target man initially on a loan deal. But Rodgers has said that they would never accept such an offer.
"You'd have to be a nutcase to accept an offer like that," he said, according to the Daily Mirror.
"There is absolutely no chance any team will do a number on us -- and certainly not Newcastle," Rodgers added.
"Newcastle got £35 million for this player last year. To even consider wanting to take him on loan is a liberty, really."