"Islamabad is not establishing any kind of relations with Israel," Alvi told the media before his departure for Turkey on a three-day official visit in the wee hours of Sunday.
It all started when an Israeli journalist Avi Scharf tweeted on October 25 that an Israeli business jet flew from Tel Aviv to Islamabad where it was on the ground for 10 hours, before flying back to Tel Aviv.
Social media users took note and flayed the Pakistan government with the Opposition also joining in to demand an explanation about the "secret Israeli mission".
Foreign Minister Shah Mehmood Qureshi also dismissed reports of an Israeli aircraft landing in Pakistan as fake and baseless.
Qureshi said that something which is not even real does not warrant a response.
Information Minister Fawad Chaudhry has also rejected the reports and said the "government would not negotiate in secret with either Israel or India".
He regretted such reports despite the fact that the Civil Aviation Authority (CAA) had also denied reports of the presence of any Israeli aircraft in Pakistani airspace or at its airports.
Pakistan Muslim League-Nawaz leader Ahsan Iqbal, who earlier led the Opposition's barrage, said Chaudhry's reaction showed the government was trying to hide something.
"The parliament should be taken into confidence over the issue," he demanded.
BBC Urdu reported that the aircraft in question was a Canadian-manufactured Bombardier Global Express with the serial number 9394. It was registered on February 22, 2017 in the Isle of Man in the UK by a company called Multibird Overseas Ltd.
The Israeli journalist later said he was not "100 per cent sure" if the plane had landed in Islamabad.
Pakistan and Israel do not have diplomatic relations and their aircraft are not allowed to use each other's airspace.