Brazilians president Patrice Motsepe turned down a bid of roughly R14.5 million from Israeli club Maccabi Haifa last week and refused to let him go for a trial at Dynamo Kiev.
The Saturday Star quoted Mphela's agent Glyn Binkin as saying over the weekend: "I can understand if they are rejecting the offer (from Haifa).
"But then let him go on trial (at Kiev), like (Knowledge) Musona went to Hoffenheim and (Morgan) Gould to England.
"They shouldn't have an issue if they are saying he no longer figures in their plans."
But the PSL outfit's stance has changed after their official website confirmed they will allow the Bafana Bafana striker to go to Parkhead in the hope he will impress their management.
Mphela is now set to depart for Scotland for a trial from 25 to 29 July. Motsepe is believed to be holding out for a fee in excess of R17 m if he is to let go of the player.
Binkin added: "I can confirm that Mphela has an offer for a trial with Glasgow Celtic. I passed the letter of invitation on to Sundowns in the last 24 hours.
"Maccabi's best offer was R14.5million (£1.3m) but there are other clubs, like Dynamo and Celtic, who could afford as much as R17 million (£1.5m).
"Clubs like that want to be able to see the player before signing, though. Katlego's an unknown quantity in Europe.
"He played there before but didn't quite make it the first time, so clubs want him to come on trial first."