Richmond Park - Town's ground - does not meet UEFA standards which forced them to make a long trip north to use Newtown's Latham Park ground against Longford Town in the first qualifying round.
However, to say Copenhagen are a bigger side would be an understatement, with the club's average home attendance over 20,000, while they have a huge following away from home.
Swansea's ground is the nearest large capacity venue capable of meeting their requirements to Carmarthen and would be available during the week of Carmarthen's European tie.
Carmarthen boss said: ''We want a ground that has the capacity to holding at least three or 4,000 people.
''It would be great if we could play the match at Swansea - it would be a privilege."
A decision should be made soon on whether Carmarthen can host their home tie at the New Stadium.