The defender has admirers from elsewhere in Europe, with the Scottish club having to reject a loan with an option-to-buy offer from Italian side Udinese in January.
Ligue 1 outfit Toulouse burst onto the scene this summer and went on to make an enquiry with Celtic about the availability of the defender.
An official bid has now been submitted that included a season-long loan deal with an option to make the move permanent next summer for a fee in the region of £3.5m.
Toulouse were also keen on including add-ons in the deal, though Celtic have now gone on to reject the offer.
Celtic manager Ange Postecoglou sees the Welsh as a part of his first-team plans and therefore does not want to let him go.
It now remains to be seen whether Toulouse return with a fresh offer or turn their attention elsewhere.
Welsh is a product of Celtic’s youth academy and has represented his club on 46 occasions since his debut in 2020.
