Norris is the reserve driver for McLaren this year, but Toro Rosso made a formal move to sign the 18-year-old on loan as a replacement for under-performing New Zealander Brendon Hartley.
Press Association reports that Toro Rosso wanted to make Norris Britain's youngest ever F1 debutant in Austria next month - just seven days before his home race at Silverstone - but McLaren are believed to have rejected their proposal.
It is believed Toro Rosso, the team with which Sebastian Vettel and more recently Max Verstappen spent their formative years in the sport, were keen for the temporary deal to be extended into next season.
McLaren's plans beyond this year, however, remain undetermined, and the British team could yet choose to promote Norris to one of their race seats in 2019.
Fernando Alonso, 36, may turn his back on McLaren, and indeed F1, at the end of the year, while there are also doubts as to whether Stoffel Vandoorne will be retained.
McLaren may also have feared risking losing their protégé on a permanent basis to their rivals at Red Bull.
A McLaren spokesperson told Press Association on Wednesday: "We are not surprised that other teams approach our drivers. They clearly believe they are as talented as we do."
Norris, from Glastonbury in Somerset, has already been earmarked as the next Lewis Hamilton after he won last year's Formula Three series before landing the reserve seat at Hamilton's former team McLaren.
The teenager holds a 27-point lead in this season's Formula Two championship, the feeder series to F1, and won impressively on his debut in Bahrain in April.
Norris is held in high regard by those at McLaren, but while he is under consideration for a full-time drive, they are wary of thrusting the youngster into the limelight prematurely.
Hamilton flourished in his first season with McLaren as a 22-year-old, but the team does not have a good record with emerging drivers in recent years.
Sergio Perez, the Mexican who replaced Hamilton, and Dane Kevin Magnussen, were both released after just one season.
Vandoorne has also struggled to keep pace with Alonso since he filled Jenson Button's boots at the beginning of last year.
McLaren have recently hired Gil de Ferran, a former IndyCar champion, to advise the team's young drivers.
McLaren will head to Montreal for this week's Canadian Grand Prix armed with a series of updates on their car, but Alonso, who will be celebrating his 300th race - becoming only the fourth driver in F1 history to reach the milestone - believes they could be in for a challenging time.
"This will certainly be a tough circuit for us, but we're making progress race by race," Alonso said.
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