Rose, 28, will be a free agent for the first time in his career. He spent last season in New York after eight years with the Chicago Bulls.
Whether the Knicks' interest ultimately leads to a contract offer for the the point guard is dependent on several factors, including the health of his surgically repaired knee and his asking price.
When asked last week about New York potentially re-signing Rose, team president Phil Jackson said "we're listening."
Rose, the 2011 NBA MVP, averaged 18 points, 4.4 assists and 3.8 rebounds in 64 games for New York, which finished 31-51.
Sources say the Knicks have discussed several different scenarios with regard to filling the void at point guard -- both in free agency and via trade. The Knicks aren't interested in paying large salaries for any free agent, which may be a sticking point in potential discussions with Rose and his agent, B.J. Armstrong.
Rose is coming off a meniscus tear in his left knee that required season-ending surgery April 5. At the time of surgery, the Knicks offered a timeline for recovery of three to six weeks.
In his season-ending news conference, Jackson said he was impressed by Rose's expressing a desire to return despite the difficult season.
"He enjoyed playing here even with the losses, which of course surprises us because he's been on some very successful teams," Jackson said. "But he wants to redeem himself as a player. Which, I like that attitude. I like who Derrick represents as himself. He's very direct about taking on a big challenge."
Both Armstrong and Rose have said in the past that Rose would be open to a return to New York and enjoyed living in relative anonymity in the city.
Rose's season in New York was not without controversy.
Members of the organization were embarrassed when he skipped a game without informing them. He later cited a family situation as his reason for leaving New York prior to the Knicks' game against the New Orleans Pelicans in early January.
It is unclear which other teams will have interest in Rose come the start of free agency Saturday. Sources in May said members of the Timberwolves viewed Rose as a potential free-agent target. It is unclear if the Timberwolves' acquisition of Rose's former teammate, Jimmy Butler, would impact a potential deal.
Some rival executives in May also tabbed the San Antonio Spurs as one team expected to have interest in Rose, depending on how the market for point guards develops. Top free agents at the position include Chris Paul, Stephen Curry, Kyle Lowry, George Hill, Jrue Holiday and Jeff Teague.
ESPN's Marc Stein reported that the Spurs are attempting to create salary-cap flexibility to pursue Paul this summer.
Rose, who has a history of serious knee injuries, said in late March that he'd prioritize winning over money as a free agent.
"Not even thinking money. I've got more than enough money saved. If I stopped playing basketball now, I'll be all right," he said. "I want to win. I want to be happy and feel at peace with myself wherever I'm at. But being at the negotiating table, you never know. I'm not going to negotiate with people where money is the No. 1 thing I'm asking for. I want to win."