According to the Portuguese, he said to the Russian billionaire: "Pay, and don't speak."
Chelsea signed Drogba from Ligue 1 side Marseille, and he went on to lead the line at Stamford Bridge for the next eight years, the Blues' most successful period in their history.
Mourinho re-signed the Ivorian in his second spell in charge at Chelsea in 2014, but he explained that first time around he was questioned about Drogba by Abramovich, per beIN Sports (h/t Tom Doyle of the Evening Standard):
"Drogba is the kind of guy who is so loyal that he never forgot that I took him from Marseille to the Premier League, where he was not even expected, because he didn't start very young at that level. He had already played for Guingamp, Marseille, Le Mans, so when I took him to Chelsea, I remember clearly Abramovich was asking me, 'Who? Who do you want as a striker?'
"With all the big names in Europe at that time, I said Drogba. ‘Who is he? Where's he playing?' [I said] 'Mr Abramovich—pay. Pay, and don't speak.'"
Chelsea paid £24 million to sign Drogba in July 2004, and in his debut season, Mourinho led the Blues to their first-ever Premier League title.
Drogba would subsequently win three more league titles with Chelsea, as well as four FA Cups, three League Cups and the 2011-12 UEFA Champions League, when he scored the winning penalty in the final.
His 157 goals make him the fourth most prolific scorer in the club's history, and he was a key figure in the spine of arguably Chelsea's best-ever team alongside Frank Lampard, Claude Makelele, John Terry and Petr Cech.
Mourinho did not pluck Drogba from complete obscurity as he won Ligue 1's Player of the Year award in 2003-04.
But he was still a relatively surprising addition, and clearly Abramovich was not completely aware of who he was spending his money on.
Drogba lived up to his fee and more, though, playing a key role in numerous Chelsea triumphs as they established themselves among Europe's superpowers following Abramovich's purchase of the club in 2003.