Conte is widely expected to leave Stamford Bridge at the end of the season after presiding over a tumultuous campaign, both on and off the field.
Despite being champions last term, Chelsea have not challenged for the title this year and look set to miss out on Champions League football as well.
Conte's willingness to criticise Chelsea's transfer policy has not helped his situation and he seems destined to leave when the season ends.
Sarri, who has done a brilliant job in Naples, has been mooted as a potential replacement, but Conte is not interested in discussing the situation.
"I think this is not my job to answer this type of question," Conte told reporters in Friday's pre-match news conference ahead of the trip to Southampton. "My job is to work with the players, finish the season as strongly as possible for us.
"As you know very well with the speculation, always there is speculation around our position, not only for me, but for a lot of coaches.
"This is our job. For me, the most important thing is to be focused on Chelsea and to try to improve this season, and finish in the best possible way.
"Honestly, I'm not curious because I don't read the papers and honestly I'm not interested in this. My only worry is to do my job, work with my players in best way possible."
Chelsea midfielder Danny Drinkwater recently suggested he will seek a move at the end of the season after starting just five Premier League games since his reported £35 million move from Leicester City in August.
"I think he's having an unlucky season because he started the season with a problem in his calf and then he continued to have problems," Conte said.
"For sure, for him this is not a lucky season, but we are talking about a good midfielder and a good player. I trust him.
"I think now is not the right moment to speak about one player, or about specifically one situation.
"I think now it's a moment we have to think about the team to try to improve our table, improve our performances and finish the season as strongly as possible."