It is thought he grew close to Kardashian, 40, when she became involved in campaigning for criminal justice reform in 2018.
Jones is the CEO of the REFORM Alliance, an initiative founded by Jay Z and Meek Mill to overhaul the justice system.
New York Post columnist Cindy Adams reported on Sunday that rumours of their romance were starting to spread.
"A new burble's coming out of the Kim Kardashian klub. Throats are clearing throats after whispering the name Van Jones," Adams wrote.
"I will not wait while everybody tells me I don't know what I'm talking about."
Jones and Kardashian, along with Ivanka Trump's husband Jared Kushner, were credited with persuading former US President Donald Trump to support the First Step Act in late 2018, which reformed tough sentencing laws.
The bill passed the Senate by an overwhelming margin of 87-12.
Jones said without Kardashian, it would never have passed. The turning point was, he said, her meeting with Trump in the Oval Office prior to the bill being introduced, when she was campaigning for the release of Alice Johnson who was serving a life sentence for a one-time drug offence.
"Let me tell you something, if Kim Kardashian had not gone to the White House and talked to Donald Trump we would not have passed this bill," Jones told TMZ at the time.
"We would not have 50,000 people fewer in federal prisons because of this bill, 100 per cent of people who are locked up in federal prisons can go home a little bit earlier if they stay out of trouble, 100 per cent of women are not going to be shackled and mistreated. Why? Because Kim Kardashian was willing to put her celebrity on the line for people she didn't even know.
"And everybody who said she was being played and she was just being used for a photo opp, how you like me now? Look at the scoreboard now – 87 votes to 12 in the US Senate. Because Kim Kardashian West was willing to stand up for somebody."
Kardashian and West are currently in the middle of divorce proceedings following seven years of marriage, though neither of them have publicly addressed the reports. They have four children together.
Meanwhile, Jones was married to producer Jana Carter for 13 years before they divorced in 2019. They have two sons.
It comes after reports Kardashian was being courted by royals, billionaires and A-list celebrities after news of her split broke.
