The 31-year-old Brazilian winger, who has been influential for Chelsea this season, is out of contract in the summer and Lampard wants to offer him a new deal.
Granovskaia, though is not so keen and is focusing on transfer targets for the summer window.
The club has a policy of only offering over-30s one-year deals, and Lampard and Granovskaia are “currently battling it out” over a decision on Willian’s future.
The tabloid claims “the Blues were ready to offer a one-year deal” but Willian revealed last month that he had been offered a two-year deal but was holding out for three years.
“Chelsea offered me two years, and they are not going to change what they offered,” Willian told Esporte Interativo after Chelsea’s defeat by Bayern Munich.
“I said I wanted three more years. The situation is difficult because of this, I really don’t know if it will be possible.
“My goal is to continue working and focus on this remainder of the season, so that Chelsea can continue winning the games for the remainder of the season.”
Lampard is a big fan of the Brazilian, who has seven goals and six assists from 37 appearances this season.
Chelsea have already secured Hakim Ziyech to play for Chelsea for next season.
Ajax star Ziyech has agreed personal terms on a five-year deal to join the Blues in the summer, with Chelsea shelling out £37million for the Morocco international.
Chelsea had pursued Ziyech hard in a bid to prise the 26-year-old away from Ajax in the January window but were rebuffed by the Amsterdam giants.