The England international is a transfer target for Drogba's club Shanghai Shenhua, and the Chinese side are poised to offer Lampard a £250,000-a-week salary.
But Chinese Super League rivals Beijin Guoan are also interested in the Chelsea star, and have the financial backing to challenge Shanghai's bid.
The 34-year-old can talk to any overseas club from January onwards as his Chelsea contract expires at the end of this season.
The Sun reports Lampard could be on his way out of Chelsea, after the Blues' owner Roman Abramovich made it clear players over the age of 30 would only be offered one-year contract extensions.
The Chinese offer is reportedly worth around £1million a month, or £36million over a three-year period.
Lampard has been at Chelsea for 11 years since moving to Stamford Bridge from West Ham in 2001.
Alongside Drogba, Lampard helped Chelsea win the Champions League and FA Cup last season, but the Ivorian striker departed for Shanghai during the summer.
Chelsea are thought to be keen to keep hold of Lampard, but the Blues star is believed to want a longer-term deal than just a one-year extension.
The England midfielder has also been linked with a move to LA Galaxy, where he would link up with former-England teammate David Beckham.