The 25-year-old, Newcastle's first signing of the January transfer window, has made 178 appearances and scored 23 goals for the French club in all competitions.
While no fee was given, the Newcastle Chronicle website reported the player had cost the Magpies 4.5 million pounds.
Saivet also played against Newcastle in both legs of a Europa League tie in 2012/13 and represented Senegal at the 2015 African Nations Cup.
"Henri is someone the club have been watching for a while, and he has developed into an excellent player at Bordeaux," manager Steve McClaren told the Newcastle website (www.nufc.com).
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"This is someone with a very good mentality, who is a tough player and a winner. Those are key qualities you need in the Premier League and we are looking forward to seeing him pulling on the Newcastle United shirt."
Newcastle are third from bottom of the Premier League, having scored just 19 goals in 20 games. Their 1-0 FA Cup third round loss to Watford on Saturday condemned the Magpies to their fourth straight defeat in all competitions.
(Reporting by Alan Baldwin, editing by Pritha Sarkar)
