The 22-year-old has been identified by the Dutch giants as their primary target to replace Lisandro Martinez, who is set to join Manchester United.
According to journalist Fabrizio Romano, a full agreement is now in place between both Rangers and Ajax over the transfer of Bassey for an initial fee worth €22m (£18.7m).
Romano adds that the deal includes a further €5m (£4.3m) in add-ons taking the total transfer package to €27m (£22.9m), which would make the defender Rangers' club-record sale.
Nathan Patterson is currently the Gers' club-record sale, after the right-back joined Everton in January for around £16m.
Rangers have also agreed to include a 10% sell-on clause in the deal for Bassey, who the Glaswegian outfit signed for just £230,000 from Leicester City in 2020.
The likes of Brighton & Hove Albion, Aston Villa and clubs in both Serie A and the Bundesliga were also understood to be interested in the Nigeria international, but Ajax have seemingly won the race for his signature.
Bassey was left out of Giovanni van Bronckhorst's squad for Rangers' 2-1 friendly victory away at Blackpool on Saturday, which sparked speculation over his future.
The versatile defender, who can operate at either centre-back or left-back, is now set to undergo his Ajax medical in the coming days before completing his switch to Amsterdam.
Bassey struggled for regular game time under former boss Steven Gerrard in his first season at Rangers, featuring in only 15 games across all competitions.
However, he became an important player for Van Bronckhorst in the 2021-22 campaign and started 45 of his 50 appearances in all tournaments, providing six assists.
Bassey excelled for Rangers in the Europa League last term and was named the Man of the Match in their heartbreaking defeat against Eintracht Frankfurt in the final.
Earlier this week, Van Bronckhorst praised Bassey's rapid rise in Glasgow over the last 12 months, and told reporters that the defender has "massive" potential which other clubs "will notice".
Bassey is set to become the second star player to leave Ibrox this summer, after attacking midfielder Joe Aribo joined Premier League outfit Southampton earlier this month for a fee which could rise to around £10m.
Rangers have already added centre-back John Souttar to their ranks on a free transfer this summer, but they will likely enter the transfer market to replace Bassey before the September 1 deadline.