The Serie A side reportedly wanted him on a six-month loan deal, with the option to buy in the summer.
Tadic's representative, Zoran Pavlovic, claims Roma's capital rivals Lazio are also targeting the 23-year-old on loan.
But the agent insists Groningen will only let the left winger go on a permanent deal.
He told Serbian publication Sport: "I have spoken to Lazio several times after Roma sent their offer to Groningen but both clubs are only interested in loaning the player, which is unacceptable for Groningen."
As well as the Serie A duo, several clubs in Germany are also thought to be targeting Tadic, with Borussia Monchengladbach believed to be among the frontrunners.
The Serbian is contracted at Groningen until 2013 and is reportedly valued at around €7.5million (£6m).
He arrived in the Eredivisie for €2m (£1.7m) in 2010 from native club Vojvodina.
