Tennis: Free tickets get British fans making late dash to Madrid
Former world number one Andy Murray had got the ball rolling after Britain's win over Germany on Friday, putting out a message on social media for fans to get in touch if they wanted a ticket for the match at La Caja Magica.
The LTA confirmed that it would honor Murray's pledge with tickets for those who responded.
Around 900 tickets have been taken, according to the BBC.
"Unfortunately, due to the incredible demand if you've not been contacted yet we won't be able to offer tickets but please do keep supporting the team at home and across social media," the LTA said on Twitter on Saturday.
Britain will need all the help they can get with world number one Rafael Nadal leading the Spanish challenge.
Murray may not even play in the tie after being left out of the last two matches -- wins over Kakakhstan and Germany.
His brother Jamie, who could play in the deciding doubles rubber if the singles are shared, said: ""All the tickets available have been taken...it's an incredible effort by you guys (the fans) and we're looking forward to seeing you."
Tickets for the semi-final were on sale from 75 euros.
The Davis Cup Finals are taking place in Madrid and are the first edition of the revamped team event in which 18 nations have battled throughout the week with the final on Sunday.
Previous Davis Cups have had a 16-team World Group with ties played at home venues throughout the year.
The change has been controversial and the week has been marked by hundreds of empty seats for the ties taking place across three courts.