Sky News has learnt that BT has appointed Lazard, the investment bank, to undertake an auction of the business, which owns 33,000 predominantly BT and Openreach-branded vehicles.
The sale process is in its early stages, but analysts suggested on Friday that a disposal could yield a price tag of £200m or more.
BT Fleet Solutions is one of several assets that the company is looking to sell as it gets ready for a handover at the top, with Philip Jansen due to replace Gavin Patterson as its chief executive in the new year.
The company is working with Credit Suisse on the sale of the Italian operations of its Global Services division, which provides IT and other services to corporate customers.
An accounting scandal in the Italian business nearly two years ago cost BT hundreds of millions of pounds and paved the way for Mr Patterson's exit.
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However, half-year results published last week offered a positive surprise to the City, with adjusted earnings before interest, tax, depreciation and amortisation up 2% to £3.7bn.
Shares in the fixed-line, mobile telephony, IT services and broadcasting group have recovered some of the ground they have lost in the recent past, and are now marginally up over the last year, giving BT a market value of just over £25bn.
This week, market rumours have suggested that Deutsche Telekom, the German telecoms giant which holds a small stake in BT following the latter's takeover of EE, is preparing a mammoth takeover bid.
People close to BT believe that such a move is unlikely at the current time.
The auction of BT Fleet Services does not only encompass the 33,000 vehicles it directly owns.
The business also serves other corporate clients, maintaining a total of 120,000 vehicles, and employs 840 in-house commercial vehicle experts.
It has a network of 65 owned garages across the UK.
A source close to BT said a disposal of the fleet arm was likely to include a leaseback arrangement to enable the company's engineers to continue to use the vehicles.
Among the challenges facing Mr Jansen when he takes the reins in February will be to convince any sceptical investors that Openreach, its semi-independent broadband infrastructure division, should remain wholly owned by BT.
Some media reports have suggested that they would be keen to see BT spin it off, but both Mr Patterson and Jan du Plessis, BT's new chairman, have signalled their continued belief in the existing ownership structure.
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In May, Mr Patterson announced plans to axe thousands of jobs alongside symbolic moves including the departure from its long-serving London headquarters, which was presented as an attempt to accelerate BT's modernisation.
BT and Lazard declined to comment on the plan to sell BT Fleet Services.