Cabrini Health, which runs a major private hospital in Malvern and smaller facilities in Brighton and Elsternwick, will move into the 2000 square metre space at 8 Redfern Street, which is off Riversdale Road near Camberwell Junction.

The five-year lease, which includes 70 car parks, was negotiated at a rate in the mid-$300s per square metre by CBRE agent Anthony Park and Gorman Commercial agent Richard Height.

Cabrini Health sold its own office building at 453 Auburn Road to Kennard's Self Storage in February. Kennards paid $12 million for the 9166 square metre building which Cabrini had bought for $2.5 million in 1996.

Shell, recently renamed Viva, occupied the office for 10 years but last December it moved into 4400 square metres of space at 720 Bourke Street in Docklands.

The Redfern Street building, which title documents indicate is owned by the Coppel entertainment family, has 8118 square metres of office space across three floors and includes a 300-bay car park.

Mr Park said negotiations were underway for an insurance company to move into the first floor and another tenant to take up the space on level two.

"The inner east is outperforming other suburban markets and the fact the Cabrini deal was completed so quickly following Shell moving out is testament to that," he said.

More than $1 billion worth of suburban property sold last year but 2015 has been marked by a paucity of supply. Institutions and private investors, who have been priced out of the hot CBD market, are looking keenly to the suburbs but existing owners are holding on to their stock.

"At the moment the residential market is so strong that many of the sites mooted for office have gone to apartment projects. There's not a lot of new office space proposed," he said.