The Red Sox traded Lester, 31, to the Oakland Athletics before the July 31 non-waiver deadline, getting outfielders Yoenis Cespedes and Jonny Gomes in return. Owner John Henry reportedly flew to Atlanta to personally talk with Lester on Friday.
Bradford reported earlier Monday that a decision from Lester is expected to come on Tuesday.
The Giants reportedly met with Levinson Sunday night in hopes of getting a deal with Lester done this week. The Cubs are believed to have offered Lester a contract worth more than $135 million last month, according to CSN Chicago.
With Lester bound for free agency after the season, the notion of him re-signing with the Red Sox was widely considered a possibility. On Friday, Jon Heyman of CBSSports.com reported that the Red Sox were "favored" to land Lester, though their offer was reportedly "maybe a bit lower" than those from other teams believed to be in the $130-140 million range. In addition to the Cubs, Giants and Red Sox, the Los Angeles Dodgers were a late entrant into the bidding for Lester.
Lester went 16-11 overall last season, posting a 2.46 ERA with 220 strikeouts and 48 walks. In the postseason, he started the American League Wild Card Game for the A's, allowing six earned runs, striking out five and walking two over 7.1 innings of a 9-8 loss to the Kansas City Royals.
- Mike Fiammetta