Gattuso was promoted from his role with the Primavera youth team to take charge of the senior side until the end of the season after Vincenzo Montella was sacked in November.
The former Milan midfielder has overseen an upturn in fortunes, with the Italian outfit unbeaten in 11 matches across all competitions following Thursday's 1-0 victory over Ludogorets, which sealed a 4-0 aggregate win.
Asked about his future after sporting director Massimiliano Mirabelli said he wanted Gattuso to remain at the club next season, the tenacious 40-year-old told Sky Sport Italia: "You have to ask the club that.
"I always give the same answers, because it's what I believe.
"It is a great honour for me to be the Milan coach. It's a lot of pressure and responsibility too, but I like that.
"I am eternally grateful to Mirabelli and Fassone for giving me this opportunity and I hope it continues."
Fabio Borini scored the only goal of the game at San Siro as Milan eased past the Bulgarian champions in the Europa League.
Patrick Cutrone teed up Borini in the first half for Milan, who made it four successive wins in all competitions.
"I did not want to concede and trusted that the defence would play well," Gattuso said post-match. "It seemed like an easy match but I wasn’t relaxed because I know these types of games can change quickly. Yet, we were on top of things and did everything not to concede.
"[Riccardo] Montolivo played his second match in three days and did very well. [Manuel] Locatelli can still give more. [Cristian] Zapata had not played since the match against Rijeka and responded positively. Also [Jose] Mauri gave a decent performance. I’m pleased with [Nikola] Kalinic. Andre Silva would have like to score but regardless of that he played a good game.
"Cutrone only needs to improve a couple of things and he could also play on the wing. Since I arrived, Borini has never given up, he plays English football, very intense and gives always his all."
Milan will find out their draw for the last 16 on Friday.