"These are purely allegations and very disappointing," Allegri told a news conference before Milan's home match against Torino on Sunday.
"It upsets me to read that I've signed for Roma. I believe myself to be an honest person and respectful towards my place of work. If anything was to happen the first person to know about it would be the president [Silvio Berlusconi] and CEO [Adriano] Galliani.
"I think this discussion is closed, and thankfully it was closed by the clubs before I had to speak."
Milan earlier denied the reports in Milan-based paper Corriere della Serra that Allegri had agreed a three-year contract with Roma after refusing a deal to join Inter Milan.
"AC Milan categorically denies what was published today in the pages of the Corriere della Sera regarding its manager Massimiliano Allegri," the club said on their website.
Roma and Inter also denied the report.
"AS Roma categorically denies today's report in Corriere della Sera... which refers to an alleged agreement made with Mr Massimiliano Allegri," Roma said.
Inter said there had been no contact between them and Allegri regarding him taking over from under-fire Andrea Stramaccioni.
"Regarding what was reported in today's edition of the Corriere della Sera on the future of the Nerazzurri bench, FC Internazionale categorically denies that has been any contact or requests for contact with AC Milan manager Massimiliano Allegri," Inter said in a statement.
Allegri's position with Milan has never been secure despite a strong second half of the season during which they have only lost one game, to leaders Juventus, and have climbed to third in Serie A.
They are a point ahead of Fiorentina and play three of their last four games against relegation strugglers Torino, Pescara and Siena, making them favourites to take the third Champions league spot.
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