Manchester United’s main stock market listing is in New York, although it has a secondary, and less widely traded, share listing in Frankfurt.
Its New York-listed shares were up 2.4% at one stage yesterday.
Mourinho, 53, is set to take over at United even though Dutchman van Gaal, who steered United to an extra-time FA Cup final win over Crystal Palace on Saturday, had a year left on his contract, according to reports.
Mourinho, sacked by Chelsea in December after his second spell in charge of the west London club, has also managed Porto, Inter Milan, and Real Madrid, winning the Champions League tournament with both Porto and Inter Milan.
Manchester United was bought in 2005 for about £800m (€1bn) by the Glazer family, owners of the Tampa Bay Buccaneers in the National Football League.
Shares in Manchester United were listed on the New York Stock Exchange in 2012 in an initial public offering.
According to Thomson Reuters data, the Glazers’ Red Football investment vehicle has a 20% stake in the club’s New York-listed shares.
US investment firms Baron Capital Management, Lansdowne Partners, Jupiter Asset Management and Fidelity are among other major institutional investors.
The New York-listed shares have fallen over 3.8% since the start of the year.
Earlier this month, Manchester United reported record third-quarter revenues and higher earnings.
However, van Gaal’s failure to steer his team to qualification for the Champions League is likely to have proved costly.
Manchester United said earlier this month that not getting into the Champions League would cost it more than £30m in revenue next season.
Manchester United have won 20 top-flight English league titles — including both the Premier League and the old Division One — more than any other club in England.
The New York shares had traded as high as $19.74 late last year, but quickly slumped after it failed to emerge from the first stage of the Champions League.