The Iceland international is on-loan at Liberty stadium from Hoffenheim.
The Sunday Mirror says United are considering a stunning £10million swoop for Sigurdsson.
United assistant Mike Phelan was sent to the Reebok Stadium last week to watch Swansea, with the champions set to face Brendan Rodgers’ side next weekend.
And he was accompanied by Martin Ferguson, the brother of United’s boss and the club’s chief scout.
His brief was to run the rule over Sigurdsson – an Icelandic international – and he was impressed with what he saw.