The Black Cats boss, who worked with the duo during his time as Manchester United manager, has revealed he is keen to work with them again at the Stadium of Light
Januzaj was handed his United debut by Moyes in 2013, but fell out of favour under Louis van Gaal and also had a disappointing loan move to Borussia Dortmund cut short last term.
Fellaini is also very familiar with the new Sunderland manager, who signed the towering Belgian for Everton and again during his time at Old Trafford.
Having replaced England boss Sam Allardyce at the Stadium of Light just 21 days before Sunderland's Premier League opener against Manchester City, Moyes is aware of the need to move fast in the transfer market as he tries to improve the Wearsiders' fortunes.
Quizzed on Januzaj and Fellaini at his first media conference since being appointed, Moyes said: "If any of those good players want to join me I'd be happy to have them because they're excellent players. We're interested.
"It does take a bit of time. They [Sunderland] only avoided relegation late in the season, we've been getting things in place and we'll make things work.
"We've got offers in for players now and I hope we can get things to move quite quickly. It's a difficult window because of the prices.
"This is now a building job. I'm here for four years I want to bring stability to the club - it reminds me of Everton.
"When I took over, Everton had been in bottom six four of the previous five years when I came in so there's a lot of similarities in a lot of ways.
"You're not going to change it in one window and if we are late we'll do the best we can. We want the right players at the right prices. We want players to do the journey with us."