Both London clubs are monitoring the Sampdoria defender, who is rated at £31m by the Serie A club.
But it has now emerged that French side Lyon have had a £16m bid for the 23-year-old centre back turned down.
Denmark defender Andersen revealed earlier this week that he is considering a move, with AC Milan also among the interested parties.
Andersen said: “There are a number of interested clubs from Europe’s biggest leagues who are discussing with my agent.
“I’ll be making a decision shortly together with my family and my agent. Maybe it’s time to move on to develop further.”
Arsenal sent scouts to watch Andersen several times this season but his asking price is currently regarded as too high for the Gunners.
And reports from Italy that Arsenal had a joint bid for Andersen and his midfield team-mate Dennis Praet turned down have been rubbished by club insiders.
Spurs regard Andersen as a potential replacement for Toby Alderweireld should the Belgium star quit the club.
Alderweireld has a clause in his contract allowing him to leave this summer for £25m but as yet there has been no serious interest in him.
Andersen is a right-footed centre-back in the same mould and started 32 Serie A games for his club this season.
The former Twente defender was set to star for Denmark in this summer’s European Under 21 Championship but was forced to pull out injured.
He has yet to earn a full cap for his country but was on the bench for their games against Kosovo and Switzerland earlier this year.