The future of France international Lloris has recently come under scrutiny, with The Mirror reporting that Paris Saint-Germain could make a move for him at the end of the season.
With Tottenham not guaranteed a place in the top four of the Premier League, PSG could lure the 28-year-old away from White Hart Lane with their riches and Champions League football.
Baumann, who has not made his international debut for Germany yet, joined Hoffenheim from Freiburg in the summer of 2014 and is rated as one of the best goalkeepers in the Bundesliga.
In 26 Bundesliga appearances so far this campaign, the 24-year-old has kept eight clean sheets and has made 2.85 saves per game on average.
While it is clear that Tottenham will need a goalkeeper if Lloris does leave at the end of the season, perhaps they could do well to promote Michel Vorm as the number one next season.
The Netherlands international is a notable Premier League custodian and has proven his quality during his time at Swansea City.
It would be wise of the Spurs chiefs to make Vorm their main goalkeeper and sign a number two capable of challenging for the top spot.