It would be Jarvis's third railway deal in little more than a year. The shares, which rose five-fold last year, edged another 5p higher to 296.5p.
With the acquisition tipped to be worth around pounds 60m-pounds 65m it would require a major rights issue, possibly of around pounds 40m. Jarvis said it hoped to make an announcement shortly.
Fastline was formed through a management buy-out 15 months ago from the former British Rail business.
Its acquisition by Jarvis would be seen a positive move for the once- struggling construction company.
Two years ago Jarvis was a struggling loss maker with a share price of just 20p. But its fortunes have been transformed since its pounds 9m acquisition last year of Northern Infrastructure (Nimco), one of seven businesses that carries out maintenance contracts for rail network operator Railtrack. Nimco has sales of around pounds 140m largely generated from guaranteed Railtrack contracts. The other infrastructure maintenance companies have been snapped up by the likes of Balfour Beatty, Amey and Amec.
Last month Jarvis paid pounds 4.9m for a half share in Relayfast, the Scottish rail engineering company. It has the rights to buy the other half for pounds 5m. Jarvis made a loss of almost pounds 5m in 1994 and profits of just pounds 500,000 the following year. In April this year broker Peel Hunt upgraded its profits forecast for the year to March from pounds 10m to pounds 14.6m.
The transformation has proved a boon for Jarvis shareholders. Last year the company's shares were the second-best performers on the stock market, rising by 506 per cent to 142.5p. The meteoric rise has continued this year, during which time they have already doubled.
The privatisation of British Rail has been fraught with controversy with the government being heavily criticised for selling the various parts too cheaply. One of the first examples of this was when Stagecoach, the bus and trains group, paid pounds 825m for Porterbrook, the train leasing company which had been bought by its management from the government for pounds 527m just eight months earlier. That deal provided huge windfall gains for directors and the buy-out's major backers, Charterhouse Bank.
Other deals have also proved lucrative. Amec paid pounds 25m for the South West Infrastructure Maintenance Company. Analysts expect that deal to provide profits of pounds 3m-pounds 4m this year.
In April Jarvis said its profits for the 15 months to March would "substantially exceed market expectations" thanks to building project gains and lower- than-expected rationalisation costs. Its projects include a contract to build a school in Dorset under the government's private finance initiative.