According to Joel Sherman of the New York Post, the squad is "contemplating if it should make a late attempt to retain Cole."
The pitcher spent the past two years with the Astros, finishing second in Cy Young voting in 2019 while going 20-5 with a league-best 2.50 ERA and 326 strikeouts.
As one of the top pitchers in the game, he has unsurprisingly been a hot commodity on the open market.
His agent, Scott Boras, explained Tuesday that the New York Yankees, Los Angeles Dodgers and Los Angeles Angels were the known bidders, per Greg Wolf of FS1. He added that there were two mystery teams.
Jon Heyman of MLB Network also reported the existence of a mystery team, but he didn't expect anyone to outbid the three big-market squads.
Sherman predicted Cole would make more than $300 million in his next deal after Stephen Strasburg set the market with $245 million on his deal.
Such a deal would put the Astros well over the luxury tax threshold, but it could be necessary to keep them a top contender after they fell short of a World Series title this past season.
Houston owner Jim Crane said in November that it's a "possibility" the organization goes over the luxury tax to re-sign Cole, per Brian McTaggart of MLB.com. He said the team was going to "take a run" at him in free agency.
The challenge will be outbidding the other interested parties around the league.