The contractor is recommended in an evaluation report due to be sent to Manchester City Council chief executive Howard Bernstein, and is set to be named preferred bidder next week.
Industry sources said Amec put in the lowest bid and scored highest in several other key categories, including technical ability and capacity to build the landmark project.
The contractor edged ahead of Bovis in the last stages of the evaluation process, partly because it had little other major work on site in Manchester city centre.
Bernstein is known to have fears about overheating in Manchester during the millennium period, and sees completing the project on time as crucial to the city's image.
Bovis too busy
Sources said Bovis was still busy with work to rebuild Manchester after the 1996 IRA bomb, although the project team was impressed with the firm's technical ability.
Laing is understood to be in third place, after the project team raised concerns about problems it experienced recently on the £120m Cardiff Millennium Stadium.
Taylor Woodrow, despite recently building several Premiership football grounds, and Mowlem, which said it would use the team that rebuilt Twickenham rugby ground, failed to make the final list.
The stadium is designed by Arup Associates and will be engineered by Ove Arup & Partners. Quantity surveyor is Poole Stokes Wood, the Manchester-based firm that recently joined forces with the world's largest QS, Davis Langdon & Everest.
These firms and the council's special projects team declined to comment, but sources said all that remains is for Bernstein to rubber-stamp the choice of Amec.
Bernstein said: "I will be receiving the evaluation report this week and I am unlikely to contradict it. I never have done before.
"I am concerned about price, quality and the ability to deliver … we do not have to look too far to see where a project like this could go wrong," he said, hinting at the huge losses on the Cardiff project.
Winning the 48 000-seat stadium would be a welcome boost for Amec, which was originally due to build an 80 000-seat stadium on the same Eastlands site for Manchester's bid to host the 2000 Olympics.
The project has been through a series of redesigns since then to meet the need for a smaller stadium for the Commonwealth Games and to ensure a sustainable future as a football ground. After the games, it will become Manchester City football club's home ground.
Alfred McAlpine was this week awarded a £3m groundworks contract for the project, and work will start this summer.
Some firms declined to bid because they believed they could lose money on the project.
However, the council is confident that a the design-and-build contract that places less risk on the contractor will ensure the project goes well.
The contractor is understood to have edged ahead of Mace, which was offering its new insurance-backed price guarantee, on the project.