United signed Turkey goalkeeper Altay Bayindir from Fenerbahce for £4.3m earlier on Friday.
Spaniard Reguilon, 26, spent last season on loan at Atletico Madrid.
"I know I can contribute to Manchester United this season," said Reguilon, who signed for Spurs in 2020.
"I am ready to fight for this group and show everyone my qualities.
"In life you have to be ready for everything and the chance to represent this great club with such an illustrious history is one that I could not turn down.
"Having spoken to the manager, I know what he needs from me, and I am ready to play my part in helping the team to achieve success."
Left-back Luke Shaw's injury has forced Erik ten Hag's side to go in search of a temporary defensive replacement.
A break clause has been included in the season-long deal for Reguilon, which can be activated in January.
Reguilon has appeared in the Premier League 52 times for Tottenham, scoring twice.
The defender has won six caps for his country and played 230 games for clubs including Real Madrid, Sevilla, Tottenham and Atletico.
He has also won the Europa League and Fifa Club World Cup.
Amrabat set to join on loan
United have also agreed a deal to sign Amrabat on loan.
After holding talks with the Serie A club, the two sides have now reached an agreement.
Amrabat is set to have a medical before signing the deal, which contains an obligation for United to buy the 27-year-old at the end of the season.
The Morocco midfielder, who starred for his country at the 2022 World Cup in Qatar where they finished fourth, has spent three seasons at La Viola, making 92 appearances.
Amrabat was left out of Fiorentina's squad for both legs of their Europa Conference League play-off against Rapid Vienna, which they won 2-1 on aggregate on Thursday.
Bayindir gives United second goalkeeper option
Bayindir, 25, who has been capped five times by his country, joins on a four-year contract.
Last season he was a key player for Fenerbahce as they won the Turkish Cup and finished second in the Super Lig.
"It is a huge honour to join Manchester United and become the first Turkish player to represent this incredible club," said Bayindir.
"I have a passion for success, and I will give everything to help this special group of players to achieve our ambitions.
"I cannot wait to start working with such an experienced goalkeeping unit. We will support each other and drive high standards every day so that each of us is ready to perform whenever called upon."
Bayindir signed for Fenerbahce from fellow Turkish side Ankaragucu in 2019, making 166 appearances and keeping 53 clean sheets.
"Altay is an excellent addition to our squad and adds further quality to our already strong group of experienced goalkeepers," said football director John Murtough.
"Having consistently performed at a high level in a strong European league, he has the qualities to support us in achieving our ambitions across all competitions this season and beyond."
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