Mourinho revealed his desire to sign a striker who isn't "afraid" to compete with Harry Kane, and Tottenham have since held talks with Eintracht Frankfurt about the possibility of signing Dost, reports David Hytner of the Guardian.
Spurs are hopeful to acquire Dost on loan, but Eintracht prefer to make any deal permanent for a player they value at €5.5 million (£5.1 million).
Tottenham lost their first match of the season two days after Mourinho said he was confident the club would add a striker who would help it deal with its congested schedule.
However, he insisted any forward brought to north London isn't expected to serve primarily as Kane's backup.
"Yes we have the best striker in the country, but we want a striker who can give us more than we have and that is to play, and why not play with Harry?" Mourinho said.
Dost, who stands at 6-foot-4, offers a wealth of experience and seems to find success wherever he goes following spells at Wolfsburg and Sporting CP.
He's Wolfsburg's fifth-highest scorer due to a stellar four-year stint, after which he moved to Portugal and finished his first season as the top scorer in the Primeira Liga with 34 goals.
Dost had eight goals and two assists in 24 Bundesliga appearances for Eintracht Frankfurt last season.