Bashir, 59, who is due to retire next year, will replace High Commissioner Shahid Malik, whose current contract is valid till the second quarter of next year.
Malik retired some time ago and his contract has been periodically renewed for six months at a stretch.
Malik could be replaced by Bashir before the expiry of the contractual period of the High Commissioner, The News daily quoted its sources as saying.
Gilani had decided that no envoy who is currently on a contract would be given further extensions.
Gilani has approved the appointment of new envoys in 14 countries.
Three more ambassadors have been asked to return to Islamabad after relinquishing their assignment so that they can be replaced with new envoys, the report said.
The daily described it as the biggest shuffle of envoys in the recent history of the Foreign Office. A formal announcement regarding the new postings and appointments is expected to be made towards the end of the week.
As part of the reshuffle, Pakistan will have new envoys in Russia, the Netherlands, Brazil, Germany, Egypt, Algeria, Cuba, Nepal, Kenya, Yemen, Tunisia, Chile and Serbia.