The announcement comes days after the MLS asked the club to settle the England star's future by the end of this week.
Milan's five million pound has already been rejected by L A Galaxy, The Sun reports.
But with Becks, 33, quickly establishing himself as an integral part of the team, Milan are willing to up their bid. They face a crucial derby match against Jose Mourinho's Inter this weekend and want Beckham's future resolved beforehand.
The player himself has made no secret of his desire to stay and praised the club for improving his fitness and game.
He said: "Milan has done me the world of good. I am much fitter than a few months ago. In a month and a half, my fitness has grown by 100 per cent. I want to stay and the negotiations are ongoing so I'm hoping for a quick outcome."&13;