Transport for London has raised its offer for around 18,000 LU staff following a dispute with union RMT.
Workers would receive a £350 flat-rate bonus, plus an extra £20 per shift and a performance-related payout.
RMT general secretary Bob Crow said the offer was a “massive shift”, which the union will consider before making a formal response.
In January, the RMT rejected a £100 bonus offer as “derisory”, pointing out that it had secured much better deals with other transport firms.
Network Rail has promised £500 to its staff and London Overground is shelling out £600 per person.
Separately, the RMT said it was balloting members at Tube Lines, which carries out maintenance work on three routes and employs 3,500 people, about industrial action over pension and travel pass rights.
The news came as Boris Johnson pledged to help London Underground cut delays by 30 per cent, through a number of schemes including weekly “war room” meetings between the heads of each Tube line and more medical training for transport police.