A number of teams are said to be interested in signing the 22-year-old this summer.
The Seagulls want a figure in the region of £40m despite him never having played a Premier League game.
Liverpool have also been linked with the defender following their sale of Dejan Lovren to Zenit St Petersburg.
The Daily Star claim Chelsea are not put off by the figure being quoted by Brighton as Lampard looks to overhaul his squad this summer.
Antonio Rudiger could reportedly be axed to fund a move for White after the Germany international has failed to convince the Blues boss.
Brighton may be reluctant to sell White, though, considering his potential at the top level.
Jason Cundy disappointed Willian is leaving Chelsea
When asked if he was ready for the top level, manager Graham Potter said: “Absolutely, 100 per cent.
“He’s our player, he’s got a long contract with us. I have known Ben for a while, I have been aware of him, wanted to try and sign him when we were at Swansea.
“I know about his qualities. He’s a fantastic player. He’s done brilliantly at Leeds and helped them get promotion – that’s fantastic for them, fantastic for him – but he’s our player and we’re looking forward to working with him.
“He complements us really well. He’s got attributes that are really, really useful and he’s going to be a top player.”