Billy Allen, head of McAlpine's business services division, said: "We're talking to people like Rangers [the team that plays at Ibrox]. We want a trophy club."
If Stiell wins the Hampden Park and Ibrox contracts it will have made a clean sweep of the biggest Glasgow stadiums. Last year it won a five-year contract with Celtic at Celtic Park.
Stiell has also pitched for 10-year contracts with BAA South-east to provide facilities services and maintenance on Heathrow, Gatwick, Stansted and Southampton airports. The contract would be worth from £2-20m a year.
Stiell is also thought to be close to deals with Manchester airport and Prestwick airport in South Ayrshire, which would be valued at £1-2m a year.
It is thought to be in exclusive negotiations with officials at Prestwick. Allen said: "We're looking at the rest of BAA's portfolio in the South as well as Manchester and Prestwick airports."
Stiell has a £5m-a-year contract to work on BAA's Scottish airports and is also involved in construction work at Heathrow's Terminal 5.
n Construction managers PCM and Mace are understood to have been shortlisted for the contracts to build a £60m headquarters for pharmaceutical firm Roche in Welwyn Garden City, Hertfordshire.
The companies prequalified along with Bovis Lend Lease in December and submitted bids in January. Bovis Lend Lease is thought to have been eliminated later.
The scheme will house 600 staff and include space for product development, sales and marketing and corporate developments.
Roche is planning to complete the project in 2005.