Moyes, appearing before the media for the first time since replacing England boss Sam Allardyce as Black Cats manager, said he was moving quickly to bring reinforcements to the Stadium of Light and is eyeing some familiar faces.
Fellaini is a long-time favourite of Moyes, who brought him to Everton in 2008 and signed him again at the start of his ill-fated stint with United.
And although he may prove out of Sunderland’s reach, out-of-favour winger Januzaj looks a realistic target. “If any of those good players want to join me I’d be happy to have them,” said Moyes.
“They’re excellent players. We’re interested but I’m sure there’s a load of other clubs that are as well.
“We’ve got offers in for people already. We’re interested in lots of good players.
“But you have to have a realistic view. There’ll be some players we can attract just now and some players we can’t.
“We’ll try and get the ones we can and the ones who want to come here and start on the new journey we’re on.”
Sunderland have yet to make a single addition to their first-team squad since narrowly avoiding relegation from the Premier League, with departures and injuries leaving them worryingly light on numbers with less than a fortnight to go before the start of the new season.
Moyes thinks that is partially down to the Football Association’s pursuit of Allardyce but vowed to bring with him the acumen he showed at Goodison Park.
His troubled 10-month spell at Old Trafford and subsequent ill-fated move to Real Sociedad were scarcely mentioned as the Scot repeatedly drew comparisons with his successful stint on Merseyside.
“We’ll get signings in. I am confident because if you look at the transfers we had at Everton I think we had the best transfer record of any club anywhere in the Premier League. I hope to somehow replicate that in my time at Sunderland. Can I do that right away? No.
“It’s not as easy now as it was before. You’re not going to change it in one transfer window and if we are a bit late in the transfer window we’ll do the best we can.
“What we want to try and do is get the right players at the right price, like we did in the early days (at Everton). Players who we think can go the journey with us.”
Meanwhile, Ireland defender Ciaran Clark is on the verge of leaving Aston Villa after fellow Championship club Newcastle United last night activated a €7m release clause in his contract.
Despite interest from West Brom last summer, Clark chose to remain at the club where he developed through the ranks, but the five-year renewal he agreed a year ago included a provision allowing the 26-year-old leave once the agreed sum was bid.
With the clubs agreed on a fee, Clark will today meet Magpies boss Rafa Benetiz to discuss personal terms and could be available for their first game since relegation from the Premier League when they face Fulham on Friday.
Hull are desperately trying to cling on to midfielder Mohamed Diame but Newcastle have also met the Senegal international’s buy-out clause. The Tigers will offer Diame a new contract. Hull are also targeting Gianfranco Zola as the club’s new manager.
Clarke’s Ireland colleague Wes Hoolahan has signed a new contract with Norwich, keeping him at Carrow Road until at least 2018.
Midfielder Hoolahan starred for Ireland at Euro 2016, including a memorable strike to cap off a man-of-the-match performance against Sweden in the group stages.
His hot form has carried into pre-season with Norwich, scoring a long-range effort against Coventry and two cool finishes against Hannover 96 at Carrow Road.
Manager Alex Neil told the club’s website: “Wes has always been a special player for us. More often than not, he’s the one I turn to when I need someone to deliver something for me, and he never really lets me down.
“He’s a quality player. We all know his qualities. He’ll be key for us this year.”
Elsewhere Leroy Sane has travelled to England ahead of a proposed move to Premier League club Manchester City, the German winger’s side Schalke 04 have confirmed.
The 20-year-old scored eight goals and had six assists for Schalke last season as the club finished fifth in the league table. Reports suggest the deal could be worth around £30m.
Schalke themselves are closing in on a season-long loan signing of Chelsea defender Baba Rahman. The Ghana full-back has met up with the Schalke squad for pre-season training, ahead of finalising the formalities.
Rahman made 23 appearances for Chelsea last term, but has been deemed surplus to immediate requirements by new boss Antonio Conte.
Ipswich have signed of Spurs winger Grant Ward, 21 for an undisclosed fee.
Former Cameroon international Alex Song has joined Russian side Rubin Kazan after his contract was cancelled by Barcelona while Martin Montoya has joined Valencia after Barca also terminated his contract.