The Canaries acquired Chelsea's Scottish midfielder Charlie Gilmour, 20, on a season-long loan last week.
Now they want to add Bournemouth's 6ft 6in holding midfielder Billing, 25, who is valued around £10-15m, says the Mirror.
City have cash to spend after banking a club record £33m from Emiliano Buendia's sale to Aston Villa.
Gilmour joins fellow newboys Milot Rashica and Angus Gunn at Carrow Road.
Ben Gibson and Dimitris Giannoulis have also been confirmed as permanent additions after clauses in their loans were activated.
Billing would offer Norwich increased physicality in central midfield and is known by sporting director Stuart Webber from his time at Huddersfield.
The Denmark international made 34 league appearances last season with eight goals and four assists as the Cherries missed out on promotion in the play-offs.
Norwich also want to bring back Oliver Skipp from Spurs for another season on loan.
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