The consortium comprising Spanish shareholder the Cosmen family and CVC Capital Partners said in a statement it had raised its indicative cash offer for the group to 765 million pounds ($1.25 billion), and would not increase it further.
It had previously offered 450 pence a share, which was rejected by the National Express board in favour of a rights issue.
Fellow transport operator Stagecoach (SGC.L) said in a separate statement it had agreed in principle to buy National Express’s UK bus and UK rail operations from the consortium if the bid is successful, but did not intend to make its own offer for the company.
National Express shares rose 13 percent to 467 pence a share by 1043 GMT. The company operates UK rail, bus and coach networks as well as operations in Spain and North America.
(Reporting by John Bowker, Editing by Paul Hoskins)