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Spain's Cosmen family, along with private equity firm CVC, have submitted another proposal after their previous attempt was rebuffed.
The group has upped its offer from 450p to 500p, a move that resulted in National Express's share price climbing 13.1 per cent to 465.9p on the stock market yesterday.
National Express rejected the last deal after saying it has the support of investors to introduce a rights issue and fund manager at three per cent stakeholder Schroders Andy Brough told Reuters : "I still think shareholders will club together to put the money up for a rights issue ... but we will leave it to them [the board]".
A spokesperson for the transport group added that it is considering the offer.
Shares in National Express fell slightly following yesterday's gain, dropping 0.04 per cent to 464.8p at the start of trading.
