The France international has reportedly been a target for the Premier League pair, with Spurs looking for options when Emmanuel Adebayor departs for the Africa Cup of Nations and Chelsea thought to be exploring striker targets for January.
The Stamford Bridge side have been linked with a move for Atletico Madrid striker Radamel Falcao, and it is believed Roberto Di Matteo could opt for Remy if he fails in his bid to sign the Colombian international.
But Baup has insisted Remy will not be sold in January.
"In my opinion, there is no question [of a sale]," said Baup in the Daily Mirror. "He is here for the whole season."
The French striker has struggled with a series of injuries this season, but Marseille have started the campaign in fine form, and currently sit second in the Ligue 1 table, level on points with Paris St Germain.
Andre Pierre Gignac leads the goalscoring charts for Marseille but the striker faces a long injury lay-off, and Remy will likely be called upon to replace his compatriot's goals.
Chelsea could now focus on other targets, namely Falcao, as Di Matteo looks to secure some competition for Fernando Torres.
At present, the Blues only have Danny Sturridge backing-up the Spaniard, and Chelsea may look to bring in another forward come January.
Spurs face the prospect of losing Adebayor in January, but the Togolese frontman has barely featured for Andre Villas-Boas' side this season since moving from Manchester City.
Tottenham have most recently been linked with Shakhtar playmaker Willian, and Villas-Boas could turn his attentions to the Brazilian.