Saints boss Russell Martin has made Stewart his number one striking target this summer, but agreeing a deal with Sunderland has proven tough.
Now a breakthrough has come, after Saints had two bids rejected, and Stewart is travelling south for his medical.
It is claimed that Southampton will pay Sunderland a fee of £8m, plus a further £2m in add-ons.
Stewart is currently out injured and Southampton will want to conduct a thorough medical before signing off on the deal and completing it.
The striker is into the final year of his contract at Sunderland and did not want to sign a new one.
Sunderland will now be expected to use the cash to fund new signings for Tony Mowbray before the window closes.
Mowbray has been clear about his need for reinforcements to be added in the final third of the pitch.
