The Dane impressed the team during a day's running at the recent Silverstone young driver test, and is currently leading the Formula Renault 3.5 standings.
McLaren has high hopes for Magnussen and team principal Martin Whitmarsh revealed that his outfit was considering options to find him an F1 race drive in 2014.
"Yes we are looking to do just that," said Whitmarsh. "We had offered Kevin to Force India [for the young driver test], who had agreed to run him.
"But they changed their plans at the last minute. Given that he thought he was running for Force India we felt we should run him that day."
McLaren currently has a technical partnership agreement with Force India, but that deal will not be extended in to 2014. That could preclude placing Magnussen at the Silverstone-based team.
One other option could be to get Magnussen a drive at Marussia, which has ties with McLaren.
Whitmarsh said that it was highly unlikely McLaren would run him itself.
"There is a tremendous amount of expectation and pressure on being a McLaren driver," he said. "I know we have had exceptions, but generally we don't have rookies in the car and we have to try to develop drivers for the longer term.
"F1 has not been good at developing drivers. At the moment we have some really interesting programmes, but they are a few years away from having a particular impact on this team."