The initial enquiries are understood to have been exploratory, with the three clubs seeking to establish what it would take to persuade the south-coast side to sell.
Scott has indicated that he is unwilling to sign a new contract after rejecting two offers over the summer. With 18 months remaining on his current agreement, Bournemouth face the prospect of his transfer value declining as his deal moves closer to its conclusion.
Liverpool and Chelsea are understood to be the most interested in securing his signature during the January window, while Arsenal would prefer to wait until the summer. However, the possibility of a bidding contest could force the north London club to reconsider its timing.
Bournemouth remain reluctant to lose one of their leading players midway through the season. The club have realistic ambitions of qualifying for the Champions League and believe Scott is important to those objectives. The midfielder is also understood to be happy to remain on the south coast for the rest of the campaign, despite his desire to play at the highest level.
The club's negotiating position is therefore firm. Bournemouth are understood to want more than £100 million for the 23-year-old, with suggestions that an offer of £80 million would be sufficient dismissed by sources close to the club.
Their valuation reflects several factors, including the cost of finding a replacement, the £25 million invested in signing Scott from Bristol City and the sell-on arrangement with his former club. Bristol City are understood to be entitled to 20 per cent of any transfer fee Bournemouth receive.
Chelsea tested Bournemouth's resolve during the summer, making several offers reported to be in the region of £70 million. Those approaches were rejected, and the Cherries have little reason to accept a substantially lower figure in January.
Scott's situation is particularly interesting because Bournemouth have previously demonstrated that they will not be pressured into selling important players. Antoine Semenyo, for example, signed a new contract before leaving the club in January.
The midfielder has not allowed the speculation surrounding his own future to affect his performances, according to people close to the club. However, his ambition to compete at the highest level is understood to be strong, and interest from three of England's leading sides underlines his growing reputation.
Chelsea have been regarded as favourites for some time, but Liverpool are now serious contenders. Their interest is strengthened by the relationship between Scott and Liverpool head coach Andoni Iraola, who previously worked with him at Bournemouth.
Arsenal's involvement could complicate the situation further, particularly if Liverpool and Chelsea decide to pursue January deals. Tottenham and Manchester City also considered Scott during the summer but turned to alternative targets after Bournemouth made their position clear.
Scott currently earns approximately £50,000 a week, meaning a move to one of his interested clubs could bring a significant increase in wages. Bournemouth, however, will only consider a departure if the financial terms reflect the value they place on him and their own ambitions for the season.