Manchester United's main stock market listing is in New York , although it has a secondary, and less widely traded, share listing in Frankfurt.
The Premier League club's Frankfurt-listed shares briefly opened up 2.4 percent, but then slipped back to stand 2.5 percent lower.
Manchester United's shares closed up 2.4 percent in New York on May 20.
British media reported that Mourinho, 53, would take over at Manchester United even though Dutchman Louis 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.
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 Europe's premier Champions League tournament with both Porto and Inter Milan.
Manchester United's main New York-listed shares have fallen by around 6 percent so far in 2016.
Earlier in May, Manchester United reported record third-quarter revenues and higher earnings.