Frasers Group plc, which runs the store, said it will cease trading - after more than 180 years of business - on May 3, 2021.
The news is thought to threaten around 200 jobs.
The announcement was made after Frasers Group plc failed to reach an agreement with building owners Anders Polvsen to continue the lease on the building.
It comes after the Edinburgh Evening News exclusively revealed in November that Mr Povlsen, a Danish billionaire, planned to reinvent the 180-year-old building on Princes Street.
It is understood that the renovation will see a hotel, cafes and rooftop restaurant and bar replace the existing department store, alongside a potential raft of luxury shops linked to his clothing empire Bestseller.
The project’s plans include a “total restoration” of the Category A listed building’s facade including the sensitive reinstatement of lost original elements.
In a statement on Monday afternoon, a spokesperson for Fraser Group Plc said: “Despite the global pandemic, numerous lockdowns and the turbulence caused for British retail, the landlord hasn’t been able to work mutually on a fair agreement, therefore, resulting in the loss of 200 jobs and a vacant site for the foreseeable future with no immediate plans.
“Our commitment to our Frasers strategy remains but landlords and retailers need to work together in a fair manner, especially when all stores are closed.”
They added that the company would like to thank Jenners staff for their “hard work and dedication.”
Mr Povlsen bought the building, which spans sections of Princes Street and Rose Street, for a reported £50m in 2017.
Anders Krogh Vogdrup, director at Bestsellers said: “Despite the substantial rent reduction already granted to Frasers and rent-free periods to cover all lockdowns, Frasers has made the decision that it does not wish to continue in occupation.
He added: “This will see the end of the 16-year association between House of Fraser and this building, but not of the 180 years of Jenners department store.“We are in talks with retailer operators and are planning a programme of works to ensure that, when safe and able to do so, Jenners will reopen.
“Our primary goal is to see the department store returned it to its former glory; Jenners of Edinburgh is an institution and, despite the changing face of retail, it is our aspiration that Jenners will continue to be a retail store for as long as we are its stewards.”
Edinburgh City Council leader, Adam McVey said: “This is terrible news for Edinburgh, Jenners is an iconic institution in the capital and a lynchpin of our retail offer in the city centre.
“The retail sector has been hit extremely hard during this pandemic and this is yet another blow.
“As a council we’re doing a huge amount to help businesses weather these challenging times, administering over £100m in grants and supporting to highlight local businesses through the Forever Edinburgh campaign.
“This announcement will obviously have the biggest impact on the employees and we’re linking with partners on many other projects supporting those individuals and families who are most at risk of falling into financial hardship and face additional barriers to employment."