The request came following the discovery of reinforced autoclave aerated concrete (RAAC) in office blocks and a lecture theatre.
The bid for funding, which would be phased over four years, came after the trust completed work to make buildings structurally safe at a cost of around £1m.
RAAC is a building material used in some buildings to form roof planks, wall panels and sometimes floor planks between the mid-1950s and mid-1990s.
The material can be susceptible to fracture or collapse with little or no warning.
A spokesperson for North Tees and Hartlepool NHS Foundation Trust said: 'RAAC was identified within some office and residential blocks at the University Hospital of North Tees. The material was found in seven non-patient facing office blocks and in the lecture theatre – and was not found in any patient-facing areas.
'All required works to make these structurally safe have been completed, with no disruption to clinical services – at a cost of around £1m, funded from the trust's capital programme.
'To remove RAAC the Trust has submitted a bid of £4.9m to NHS England, phased over four years. We attend and update at the regular NHS England RAAC meetings.'