It is in the process of acquiring the neighbouring Land Registry building.
Earlier this year, the Land Registry confirmed it was in discussions over the sale of the building, at Parkside Court, Park Hall Way, amid plans to move to another site.
The organisation employs around 300 people in the town and is set to relocate to another premises in the borough.
Now Thomas Telford School has confirmed its bid to take on the site, which comes as it holds a consultation on increasing the annual intake of pupils, along with sixth form numbers.
The school has launched a four-week consultation on its plans.
The school’s head, Sir Kevin Satchwell, said they are working with Telford & Wrekin Council to increase pupil capacity.
He said: "Demand for places at the school is always high. To be able to offer more places to children in Telford gives us the chance to create more success stories.
"Telford is a vibrant, growing town and we are pleased to be able to work with the local authority to create the additional places that are needed.”
In a letter sent to parents last month, the school’s governing board said it wants to increase the number of admissions because of the need for more secondary school places in the town. The consultation will conclude on June 29.
Currently the school can admit 182 pupils a year, but if it is given permission for the expansion it will be able to increase this to 286. The plans would also see the total number of sixth form pupils increase from 500 to 600.
The Land Registry premises would allow substantial room for the school to expand.
In a letter to parents, Sir Kevin outlined why the school wants to increase the number of pupils.
He said: "We wish to advise that the governing board of Thomas Telford School are launching a four-week consultation regarding its intention to increase the admissions intake to the school. The consultation will commence with immediate effect and will end at noon on June 29.
"The reasons for the proposal are: The need for additional secondary school places in Telford, to offer more applicants the opportunity of gaining admission to the school, to fully utilise the additional building which is in the process of being acquired.”
HM Land Registry registers the ownership of property. It is one of the largest property databases in Europe and at the peak of the property boom in 2007, £1 million worth of property was processed every minute in England and Wales.