The out-of-contract striker's manager Edzai Kasinauyo confirms that Mutuma is at Swallows.
"He will train with them for 10 days and it is like an assessment and a closer look at him. He may get a deal as they are looking for a striker, he is an option for them," Kasinauyo tells KickOff.com.
Mutuma joined Zimbabwe's Dynamos after his contract was not renewed by Phunya Sele Sele last season.
But the player did not kick a ball in the Zimbabwean Premiership after the Premier Soccer League refused him registration on the basis that he was offloaded by Celtic after the international transfer period.
However, Mutuma had upon joining Dynamos made it clear that his intention was to return to South Africa, where his future was.
"The money is not good here and if talking of my future after football, it is in South Africa where the money is," says Mutuma.
Mutuma, who left Dynamos at the end of the 2012 soccer season when he finished as the country's top goalscorer with 14 goals, failed to recapture his Zimbabwean form at Celtic and was discarded by the club after German coach Ernst Middendorp felt he was not good enough.
Should he fail to make the grade at Swallows, Mutuma could still return to Dynamos as the second transfer period opens next month and he can be registered to play for the 1998 CAF Champions League finalists.
By Michael Kariati