Italian media have been reporting since Monday that official confirmation of the change was imminent.
Seedorf, who had no previous coaching experience when he took over, has only been in charge since January when he was given a two-and-a-half year contract.
Inzaghi, who like the Dutchman has never coached at senior level, is currently looking after Milan's under-19 team.
"I'd like to underline that I'm the Primavera (under 19) coach and will be until June 2016 and I've had no other communications regarding that," Inzaghi told reporters on Thursday, according to Milan's official website.
"I'm a club employee and obviously if they ask me to cover other roles, I'd be happy to, otherwise I'll do the job that I'm doing now."
"What I've been reading has been emotional for me. This club is my life, my history," he added.
"I stopped playing for this club and anything they ask me to do, I'll do. I could have left in January but I was happily forced to stay. Until I'm told otherwise, I'll continue to work with my players".
Inzaghi spent 11 seasons playing for Milan, where he won two Serie A titles and two Champions Leagues, and won 57 caps for Italy.
He was on the end of one of former Manchester United manager Alex Ferguson's most famous quotes when the Scot said: "That lad must have been born offside."
Seedorf has quickly fallen out of favour at Milan, according to Italian media, despite overseeing a significant improvement in results.
Milan won 11 out of 19 matches under the Dutchman, who spent 10 seasons of his playing career at the club, compared to only five of the previous nine under predecessor Massimiliano Allegri.
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