The Premier League duo are understood to have submitted offers of between 25 and 40 million euros, however, the player turned them both down.
The pressure has therefore been piled on the Nerazzurri to move to complete the signing soon, as Sturridge looks for his immediate future to be resolved.
Sturridge, 28, is linked with a move away from Anfield in January after starting just five league games this season.
"I don't want to. whatever I could say would only kill your stories or give you a story for one time". But he did not play the last few weeks for different reasons.
"We do that all the time, the only thing is with one or two players is we make a decision on is it for now or is it for the summer?"
"He lights up this team", Rodgers explained in 2015.
When Jurgen Klopp succeeded the Northern Irishman as Reds manager, the marksman's availability still dominated the agenda.
Danny Ings is being stopped from leaving as Liverpool believe Sturridge will go before Wednesday's deadline. He was still the talking point.
"In general more in than out in training, but the time he was in he was a little unlucky with games".
Lucas' career has stagnated in his recent years, following an extremely promising start and the player is hopeful of getting back on track with a move to Spurs. "There are a lot of rumours around and at this moment there is nothing to say about it. When we have something to say about it we will say something, but so far we have needed him and we will see what happens in the next few days".
A move may ultimately not materialise in this window if there is no agreement that placates the club and the player.