The forward stayed behind to complete the move, with the reported fee in the region of £50million, after the Liverpool squad left for their pre-season tour of Thailand.
Liverpool finally accepted the offer for the player over the weekend after Sterling had made no secret of his desire to leave.
Rodgers insisted there were no hard feelings between himself and the 20-year-old.
Speaking at a press conference in Bangkok, the Northern Irishman said: “I spoke to Raheem at length again before I left (on Sunday), so there is no issue and no problem.
“Two clubs have agreed a deal and subject to a medical he will move on. But at this moment in time the interest is with the players that are here.”
Sterling had reportedly asked Rodgers not to include him in the tour party and, when City tabled their improved bid on Sunday, Liverpool accepted the deal was done.
It will bring to an end an acrimonious six-month saga which began when Sterling rejected what Rodgers described as an “incredible” new contract offer in February.
Sterling went on to explain in an unsanctioned BBC interview why he had rejected the deal, before his agent Aidy Ward claimed the player would not sign a new deal, even for “£900,000 a week”.
Merseyside Police confirmed on Monday evening that they were investigating Twitter threats made towards Sterling and his three-year-old daughter after news of his imminent switch was announced.
“The language used in these tweets is abhorrent,” a Merseyside Police spokesman said in the Liverpool Echo.
“We are taking the incident extremely seriously and an investigation by detectives in Liverpool CID is ongoing.
“People who use the internet to target others and who commit criminal offences such as harassment or hate crime in doing so are not beyond the law and can expect to be brought to justice.”