Fowler, 46, is manager of East Bengal in the Indian Super League, where he was appointed last year. But BBC Radio Wiltshire reports he is now in talks with Swindon.
The Robins, relegated from League One last month after one season, have confirmed caretaker manager Tommy Wright has left the club.
Previous manager John Sheridan left in April after five months in charge.
Before moving to India, Fowler had spells in charge of Guangthong United in Thailand and Brisbane Roar in Australia.
His assistant in all three of his managerial jobs has been former Everton midfielder Tony Grant, who spent time on loan at Swindon in the 1990s.