Mourinho said that the 20-year-old Rashford would continue to feature prominently in the first team.
Daley Blind has fallen down the pecking order at Manchester United this season and has made just three starts in the Premier League so far.
Carrick returned to the United eleven in their 4-0 win over Yeovil Town in the fourth round of the FA Cup .
Juan Mata's earlier contract was set to expire after the end of the 2017/18 season. United triggered a clause in his contract that will extend his stay at Old Trafford until 2019.
Zlatan Ibrahimovic has been sidelined since Boxing Day with a recurrence of his knee injury that kept him out last year.
Max Dunne and George Tanner have become the latest players to sign their first professional contracts at Manchester United
Ibrahimovic, 36, has not featured for the Red Devils since the 2-2 draw with Burnley on 26 December 2017.
Henrikh Mkhitaryan completed a move to north London in a swap deal which saw Alexis Sanchez go the other way to Manchester United.
Sanchez, 29, joined the Red Devils on a four-and-a-half-year contract from the Gunners earlier this month.
Sanchez failed to inform UK Anti-Doping of his whereabouts on Monday while he was finalising his switch to United.
Italy international Jorginho, 26, has started 18 Serie A games for Napoli this season, scoring two goals.
Teenager Angel Gomes could make only his second senior Manchester United appearance in the FA Cup fourth-round tie against Yeovil Town at Huish Park on Friday night.
Manchester United have rejected Juventus' loan offer with obligation to buy Matteo Darmian for £19.25m
Juventus are keen to sign Matteo Darmian, who will be allowed to leave the club in the summer, but they have to meet United's £26.5m valuation.
Manchester United have been linked with a move for Nice's star midfielder Jean Michael Seri, who is also a transfer target for their neighbours Manchester City.
Jose Mourinho has committed his future to Manchester United by signing a new deal that will keep him at Old Trafford until 2020.
Axel Tuanzebe made just three appearances for the senior team in the first half of the season but Mourinho is keen for the defender to get regular first-team experience.
Paul Merson discusses Alexis Sanchez's move to Manchester United and Jose Mourinho's contract extension
Paul Merson believes Jose Mourinho's contract extension shows Manchester United can catch Manchester City going forward.
Jose Mourinho claims he is "really honoured and proud" to be the Manchester United manager after extending his stay at Old Trafford by at least another year.
Jose Mourinho says he can use Alexis Sanchez on the right, the left, up front and behind the striker following his arrival from Arsenal.
Pep Guardiola's side have had two bids rejected for the Brazilian midfielder, who reportedly prefers a move to The Etihad Stadium.
Manchester United transfer rumour: Red Devils join Inter Milan in race to sign Real Madrid's Mateo Kovacic
Mateo Kovacic signed a six-year deal with Real Madrid when he joined the Champions League winners from Inter Milan in 2015.
Jean-Michael Seri has attracted interest from Manchester United and Manchester City. Jose Mourinho and Pep Guardiola's side are set to go head-to-head in January.
How will Manchester United line up with new signing Alexis Sanchez and who is at risk of losing their place in the starting XI?
Alviol had recommended Sanchez to Mourinho again in 2013 during his second spell at Chelsea but the Chilean was already at Barcelona.
Arsenal hopeful Henrikh Mkhitaryan and Aubameyang arrival will convince Ozil to extend Emirates stay
Mesut Ozil has less than six months left on his contract at Arsenal and is attracting interest from Manchester United, Inter Milan and Juventus.
Eric Bailly is behind schedule in his bid to return from ankle surgery after his Manchester United return was pencilled in for the Champions League trip to Sevilla.
Jose Mourinho to target midfield reinforcements following Alexis Sanchez arrival at Manchester United
Manchester United will now shift their focus to signing midfielders following the arrival of Alexis Sanchez and are 'confident' over deals for Julian Weigl and Carlos Soler.
Manchester City transfer rumour: Shakhtar Donetsk midfielder Fred agrees to join Pep Guardiola's side
Unlike in the case of Alexis Sanchez, Manchester City have managed to get ahead of Manchester United in signing Fred from Shakthar Donetsk.
Alexis Sanchez became Manchester United's first signing of the January transfer window. The Chilean international has been handed the iconic No 7 shirt at Old Trafford.
Sanchez has signed a four-and-a-half year contract at Old Trafford worth reportedly around £500,000-a-week while Mkhitaryan has penned a long-term contract with Wenger's side.