The 25-year-old Senegal international has joined the club for an undisclosed fee, understood to be in the region of £5million, on a five and a half-year contract.
Saivet said: "I am very happy to be here at Newcastle. I am really proud to be able to play for this club, and I am really excited at the challenge that lies ahead.
"St James' Park is a really impressive and special stadium. In France, there is nothing that compares to this, and I can't wait to get out there with my new team-mates and help the team climb the table."
Saivet's arrival will come as a boost to McClaren, whose side currently sits in 18th place in the Barclays Premier League table and has lost each of its last four games in all competitions 1-0.
The former England boss will hope his acquisition, who started his career as a winger, but has lined up in central midfield for Bordeaux this season, is the first of several to arrive at St James' this month as he battles against relegation.
The 54-year-old said: "Henri is someone the club has been watching for a while, and he has developed into an excellent player at Bordeaux.
"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."
Saivet came through the ranks at Bordeaux, making his senior debut at the age of 17 and going on to score 23 goals in 178 appearances.