The USA international is out of contract at Fulham.
Hyndman is out of contract at Craven Cottage and had been linked with a move to Parkhead, while Borussia Dortmund and AC Milan are also listed as suitors.
However, the USA international is now said to be Bournemouth bound, with the finishing touches being applied to a long-term deal.
Hyndman tracks Chelsea's Ruben Loftus-Cheek during an under-21 fixture between the USA and England
The Sun reports that Cherries boss Eddie Howe is planning to include Hyndman in his first-team plans next season - an incentive that Celtic, whose squad contains an abundance of central-midfielders, would have been unlikely to offer.
Hyndman, for whom Bournemouth will be required to pay compensation, made his Fulham debut upon the club's relegation to the Championship in 2014.
The 20-year-old went on to make 12 first-team appearances during the 2014-15 campaign, with a further 16 following last season.
Hyndman was brought back to the fore by Fulham boss Slavisa Jokanovic
Should Hyndman complete a move to the Vitality Stadium, he would become the Cherries' first close-season signing.
However, the report claims defender Tommy Elphick could be sold to Aston Villa, though only 'for the right price'.
Elphick could depart the Vitality Stadium