The couple has been dating on and off since 2017 and, according to Britain's The Sun newspaper, the pair have grown "extremely close" while in lockdown together in Los Angeles during the Covid-19 crisis.
A source told the publication, "Chris and Dakota have had a great summer and it has really cemented their relationship."
"They've had ups and downs in the past, and although Chris was worried they wanted different things, Dakota is super-happy and ready to commit," the insider added. "Chris has bought her a ring as a sign of his intent, and no one would be surprised if an engagement announcement followed."
Chris was previously married to actress Gwyneth Paltrow for 13 years, and shares his two children - 16-year-old daughter Apple and 14-year-old son Moses - with her.
While the "Avengers: Endgame" actress is now married to TV mogul Brad Falchuk, the former couple has remained close and Chris and Dakota often vacation with Gwyneth and Brad, as they spend time with their kids.