The 23-year-old has seen his first team opportunities being limited at the Santiago Bernabeu this season. He has made only one start in the league, which came in his side's 3-0 defeat at the hands of Barcelona on 23 December.
According to the Italian publication Calciomercato, Kovacic is keen on making it to the Croatia squad for the summer's World Cup in Russia. The only way he can improve his chances is by featuring regularly in the second-half of the season.
The Croatian international suffered a muscular problem in August and had another setback in September. He was forced off the pitch during the Champions League fixture after having partially torn an adductor muscle in his right leg. He returned to full fitness in late November.
Since then, Kovacic has been forced to settle for a place on Real's bench. He has been left frustrated with the lack of playing time under Zinedine Zidane in recent weeks.
Jose Mourinho currently has the services of Paul Pogba, last summer signing Nemanja Matic, Ander Herrera, Marouane Fellaini and Michael Carrick as the recognised midfielders in the squad.
Carrick will retire after the end of the season and join United's coaching staff from the next campaign. In addition to this, the Belgium international's contract at Old Trafford will run down in June. If Fellaini decides against signing a new deal, the Red Devils will lose two midfielders in the summer.
United are looking to bolster their midfield and have identified Kovacic as a transfer target. Any move from the 20-time English champions in signing the Real midfielder will see them face competition from Inter Milan.
Kovacic's former club are also interested in signing the player, just after two seasons after allowing him to leave San Siro for €35m (£30.5m, $43.3m) in 2015. Inter are now looking at the option of bringing the Croatian midfielder back to the club.
Both United and the Serie A outfit are set to face an uphill task as Real general director Jose Angel Sanchez revealed the Champions League winners will not allow Kovacic to leave in January. He also confirmed the La Liga giants rejected multiple approaches from top clubs across Europe for their player's signature last summer.
"Kovacic is not for sale, we will never sell him! If there are any offers for him in January, we will reject them. Last summer there were a lot of offers and questions about him," Sanchez told Sportske Novosti in November, as quoted by Goal.com.
"Juventus and AC Milan didn't send offers, but Inter must have sent one. There was one from England, it was Tottenham."