The 29-year-old Serb, who joined for £40m, was United’s third summer buy after Benfica defender Victor Lindelof and Everton striker Romelu Lukaku.
Chelsea won the Premier League title last season, finishing 24 points clear of sixth-placed United.
“Right now Chelsea are in a worse position than last year,” added Wright.
As well as selling Matic, Chelsea have told Spain striker Diego Costa he is free to leave, decided not to offer a new contract to veteran skipper John Terry and sold back-up keeper Asmir Begovic to Bournemouth.
The Blues have signed Monaco midfielder Tiemoue Bakayoko, Real Madrid striker Alvaro Morata, Roma defender Antonio Rudiger and former Manchester City keeper Willy Caballero.
“They have signed Bakayoko from Monaco to replace Matic, but he has to come in and hit the ground running because Chelsea want to go far in the Champions League and defend the Premier League trophy,” Wright told BBC Radio 5 live’s Monday Night Club.
“They are getting rid of players who are experienced and from who you know exactly what you’re going to get.”
‘Chelsea may struggle to replace Costa goals’
Costa, 28, scored 20 goals in 35 Premier League games to help the Blues win the title but has been told by manager Antonio Conte he is free to leave Stamford Bridge.
The former Atletico Madrid striker was dropped in January after a dispute with a fitness coach, amid rumours of a move to China, although Conte said he was “very happy to play” when recalled.
Chelsea completed the club record £60m signing of Spain striker Morata, 24, on a five-year deal last month.
“With Costa not involved you have taken away 20 goals and someone has to replicate that,” said Wright, who scored 185 goals in 288 games for Arsenal between 1991 and 1998.
“I’m not saying Morata won’t do that, but when you look at Chelsea as a whole, they have a good team but are lacking depth.”
Conte is being undermined – Sutton
After winning the Premier League title in his debut campaign, Conte says the new season will be “the most difficult” of his career.
The Italian, 48, has spoken of the need for Chelsea to sign more players and gave terse answers when questioned about the matter again in his news conference after Sunday’s Community Shield defeat by Arsenal.
The former Juventus and Italy boss declined to answer when asked if his squad was ready to challenge in all four competitions this season.
Last week he also appeared to indicate he did not want Matic to leave.
“He is being undermined. To sell one of your best players to a rival? If he didn’t want him to go and he has been sold then it is a problem,” former Chelsea striker Chris Sutton said.
“He is extremely agitated. If he doesn’t get players in then it wouldn’t surprise me if he walked.”