After the conversation, Jackson told people within the Knicks organization that he believed Walton would stay with the Warriors, a source told ESPN's Chris Broussard.
Jackson said Thursday that he planned to talk to multiple candidates about the vacancy. In addition to Walton, interim head coach Kurt Rambis is under consideration.
"People I probably know will be in the interview process," Jackson said. "I will reach out to make connections to some people."
On Monday, Jackson responded to a media report that Walton had told the Knicks he wasn't interested in the job. Jackson tweeted that no one has been offered or turned down the Knicks' head-coaching position yet.
It is unclear if Walton is looking to leave his position with Golden State or if he has interest in the Knicks' vacancy. A source said Walton hasn't made any firm decisions.
Walton played for Jackson while Jackson coached the Los Angeles Lakers. He has an intimate knowledge of the triangle offense, which Jackson ran in Los Angeles and has implemented in New York.
The Knicks will likely be among a group of teams pursuing Walton this offseason. New York is in no rush to make a hire, a source told Broussard, because the team has zero picks in this year's draft.
With Walton serving as interim coach, the Warriors opened the season with an NBA-record 24 wins. They went 39-4 under Walton before head coach Steve Kerr returned from a leave of absence due to a back ailment. The NBA has given Kerr credit for the wins, but opposing teams noticed how well Walton handled things on the bench.
Jackson's recent talks with Walton were first reported by the New York Post.
Walton's father, Hall of Fame center and ESPN broadcaster Bill Walton, said his son should stick with Golden State.
"It doesn't get any better than what he's got right now -- ever in life -- and money will not make that happen again," Walton said last month on ESPN's First Take. "It's there now. Head-coaching jobs, they're open for a reason, and what he's got -- just stay there."
Sources told ESPN.com last week that the Knicks are interested in former Cavaliers coach David Blatt, among others. Rambis, who went 9-19 as the Knicks' interim coach, remains a strong candidate for the full-time job.