"I want to make it absolutely clear - I will not quit even if we lose to Basel," he told reporters on Wednesday ahead of Thursday's Europa League last-16, second-leg tie against the Swiss side at St Petersburg's Petrovsky stadium.
The Russian champions must overturn a 2-0 defeat from last week's first leg. It was Zenit's third consecutive defeat in all competitions and pundits have led calls for the Italian's exit.
"I think Spalletti has exhausted himself at Zenit. You can see it by just looking at him - it seems he just doesn't have the fire anymore," former Zenit and Russia striker Alexander Panov was quoted as saying in Tuesday's Sport-Express daily.
Zenit lost to Liverpool 3-1 in last month's Europa League last-32 tie but scraped through to the next round on the away goal rule having won the first leg 2-0 at home a week earlier.
They were soundly beaten by Basel in the last-16, first-leg encounter before going down to Rubin Kazan 1-0 at the weekend when the Russian league season resumed after a three-month winter break.
"Of course, if you don't get results your role as a coach is in doubt. It's a normal thing in football," said Spalletti, who has led Zenit to back-to-back Russian league titles in the last two years.
The Kazan defeat meant Zenit dropped eight points behind league leaders CSKA Moscow with 10 games left in the season.
"I'm not a quitter, I know that we can change this situation around," said Spalletti, adding that first-choice goalkeeper Vyacheslav Malafeyev will miss Thursday's match against Basel with a back injury.
Zenit will also be without Italy defender Domenico Criscito, who is due to have surgery to repair torn knee ligaments.
Media reports have linked fellow Manchester City manager Roberto Mancini with the job at big spenders Zenit, bankrolled by Russian energy giant Gazprom.
Zenit, who endured a period of turmoil earlier in the season with several senior players unhappy after they splashed out more than $100 million on Brazil striker Hulk and Belgium midfielder Axel Witsel, have denied the reports about Mancini.