The former Blackpool midfielder has since signed a fresh three-and-a-half year deal with the club, keeping him at the County Ground until 2023.
It comes after Oxford United turned down Blackpool's request to speak to their manager Karl Robinson earlier in the week.
"Earlier this week, club chairman Lee Power received an approach from Blackpool for their vacant managerial position; however, with contract discussions already well underway, this was rejected by the club," Swindon said in a statement.
"We are thrilled that Richie has decided to continue his tenure and excited by the challenge of continuing the upward trajectory that the club is on with him at the helm.
"Since taking over, Richie has taken the team from 18th in League Two to the top of the division, while playing some of the most attractive and intense football the County Ground has seen for a long time.
"Wellens’ on-pitch success has been mirrored by his endeavours within the community and his desire to give the people of Swindon and Wiltshire a club they can be proud of."
Wellens, who played for Blackpool between 2000 and 2005, currently has Swindon sitting top of the League Two table.
The 39-year-old was interviewed for the Blackpool job prior to Terry McPhillips taking over in September 2018.
He was later appointed Swindon boss in November of that year.
Swindon chairman Lee Power said: “We’re thrilled that Richie’s committed his future to the football club.
"He’s a young manager, still, with huge potential and we’re really excited by the way he has got the team playing and the results that have come with that.
“His hard work on the training pitch, and out watching games, is second to none as well as his efforts around the club and in the community.
"It really was our top priority to get this sorted as early as possible and I’m really glad to have got it all over the line so that Richie and the club as a whole can concentrate on the aim of promotion.”