The 24-year-old Mexican international has played just one game for Andoni Iraola's side this season.
With the January transfer window set to open later this week, Wilfried Nancy will be looking to strengthen his team ahead of the second half of the season.
Nancy's side is short of quality options on the right, following Alistair Johnston's second hamstring injury this season.
The defender has featured in matchday squads and made 13 appearances for the Cherries last season, but has struggled to make it on to the pitch this time out.
According to new reports, the English Premier League club are likely to sanction a loan deal for the 24-year-old, who came through the ranks at LA Galaxy. The right-back then made a move to Barcelona in 2023.
He spent a year on the books of the Spanish giants, playing on loan at Las Palmas. His performances in La Liga won him a transfer to Bournemouth the following season.
The defender will be looking to get guaranteed game time, ahead of Mexico's World Cup campaign as joint-hosts next year, alongside the USA and Canada, respectively.
He currently has 16 caps for his national team, after pledging his footballing allegiance to Mexico over the United States.
For the United States, he made one friendly appearance in 2020, before making the international switch the following year.
Nancy will also be familiar with the player as a manager, having come up against the defender during his time in the MLS with CF Montreal and Columbus Crew.
Should Araujo make the move to Parkhead, he will be competing with the current defensive options on the right: Anthony Ralston and Colby Donovan, with Yang Hyun-jun playing further forward as a wing-back.