The US private equity firm has picked investment bank Lazard as financial adviser for a potential London listing, the sources said.
Lone Star is also talking to underwriting banks about mandates, the sources said.
No final decisions have been made and Lone Star may decide to hold on to the business for longer.
MRH is one of the largest independent service station owners and operators in Britain, according to its website.
It had acquired Kuwait Petroleum GB almost 15 years ago and subsequently bought outlets from Texaco, BP and Esso.
It said it now has over 450 fuel forecourts, most of which operate under the Esso and BP names.
Like Applegreen, the Dublin-listed forecourt operator, MRH also offers food and beverages with Costa Coffee and Subway.
European companies are flocking to list this quarter to capitalise on investor appetite after political uncertainty subsides in key markets including Germany, France, and the UK.
TI Fluid Systems, a UK automotive-parts company owned by Bain Capital, and Novares, a supplier of plastic components for cars owned by Equistone Partners Europe, have announced IPO plans.
Investors have been increasingly focusing on assets such as petrol stations, attracted by stable and more predictable returns.
Abu Dhabi National Oil is preparing an IPO of its service stations unit that could value the business at as much as $14bn (€11.9bn) sources said in July.
Lone Star completed the purchase of MRH in January last year for an undisclosed amount.
In 2015, buyout firm TDR Capital took a minority stake in rival Euro Garages, one of the largest independent forecourt operators in the U.K. with a portfolio of over 340 sites.
Lone Star runs 17 private equity funds with total capital commitments of over $70bn. Companies have priced $44.8bn of IPOs this year in Europe, according to data, up from $32.4bn in the same period last year.
Among other deals in the pipeline for this year, SMCP, the owner of French fashion chains Sandro, Maje and Claudie Pierlot, said it plans to sell at least €517m of stock. In Italy, Pirelli went public last week.
Bloomberg and Irish