Goldman now has a 140p target price on FirstGroup, which operates buses in Glasgow and Aberdeen, up from 135p. It rates it a "buy".
The bank has cut Stagecoach, which is chaired by co-founder Sir Brian Souter, from "neutral" to "sell".
The rating is a boost to FirstGroup, which has had a tumultuous year including a £615 million rights issue to bolster its balance sheet which was accompanied by the announcement that long-standing chairman Martin Gilbert was to leave.
Last week, it faced demands from activist investor Sandell Asset Management, the owner of 3.1% of the company, that it offload its US coach service Greyhound and float on the stock exchange the rest of its North American arm.
Meanwhile Stagecoach, which operates the West Coast Mainline as part of Virgin Rail, delighted the City with an 11% hike in its dividend on the back of strong trading in the United States and higher profits.
FirstGroup's shares have changed hands for as much as 183p over the last year, but dropped as low as 90.3p as it sought cash from investors.
Last night the stock closed down 0.8p or 0.67% at 118.5p, still well adrift of Goldman's target. Stagecoach closed down 4.6p or 1.25% at 365p. This is up a fifth in 2013 to date even after coming off a high of 376.5p set last week.
FirstGroup, run by former London Underground chief Tim O'Toole, has long had a larger stock market worth than its rival. However, Stagecoach overtook it in 2012. It is now worth £2.1 billion versus £1.4bn for FirstGroup.