City councillors backed the plan - which could deliver a Â£3.3bn boost to the regional economy - on Thursday.
The Swansea Bay City Region would boost digital technology for advances in areas such as healthcare and energy.
Swansea University is carrying out a statistical analysis of what the deal could be worth to the area.
It believes it would deliver an estimated Â£3.3bn boost to the regional economy over 15 years, generating over 9,500 direct new jobs.
About 33,000 jobs in total could be created across the region when spin-offs, relocations, construction jobs and business opportunities for local suppliers are taken into account, it believes.
Vice chancellor Richard Davies said it would be a "catalyst" for the wider economy and show the "ambition" the area has.
He added: "Our world-leading research, in collaboration with industry, will continue to grow and have a real impact."
Figures behind the plan will soon visit London to press the UK government to approve the scheme.
Reports have also gone to other councils involved -  Carmarthenshire, Neath Port Talbot and Pembrokeshire, for approval.
Approval for the Swansea Bay City Region deal would be a "catalyst" and show the area's "ambition", according to Swansea University.