The series in New Zealand will be crucial to India's hopes of reaching next year's World Test Championship (WTC) final, with every game now a must-win for the Shubman Gill-led side.
Hampshire said the decision was taken following "feedback from our staff and supporters", given the player's controversial past.
India A will tour New Zealand for a three-match four-day red-ball series against New Zealand A from October 24 to November 10.
India were left frustrated by Sonal Dinusha's gritty century as the 2nd Test match against Sri Lanka ended in a draw in Colombo on Thursday.
Jasprit Bumrah's latest reign at the top lasted just two months but he has a combined 697 days as number one.
Pakistan slumped to an innings and 103-run loss against England in the first Test match at Headingley on Friday.
Brendon McCullum has hailed his successor Stephen Fleming as "like me, but better", after England appointed his former New Zealand teammate as Test coach.
Mitchell Santner was given not out by the umpires, but himself asked for a third umpire check that ultimately led to his dismissal.
Former India wicketkeeper-batter Saba Karim believes India's batting unit emerged with plenty of positives from the ODI series against England.
Mark Chapman and Mitchell Santner guided New Zealand to a tense one-wicket win over West Indies in the fourth one-day cricket international Sunday and to an unassailable 3-1 lead in the five-match series.
Left-arm spinner Jayden Lennox took 5-19 as West Indies collapsed from 63-0 to be all out for 138 and New Zealand went on to win the second ODI on Monday by five wickets.
Keacy Carty was dismissed on 95 and Shai Hope made an unbeaten 87 in a 131-run third wicket partnership which steered the West Indies to a seven wicket win over New Zealand on Saturday in the first of five ODIs.
Former New Zealand cricketer Ross Taylor hailed Prime Minister Narendra Modi's landmark visit to the island nation, describing it as an "absolutely huge" moment for the bilateral relationship between the two nations.
The England and Wales Cricket Board (ECB) was contacted by the International Cricket Council (ICC) over Ben Stokes' retirement video during the third Test match against New Zealand at Trent Bridge, according to a report.
The star will play the upcoming five-match ODI series against the West Indies after sitting out New Zealand's 2-1 Test series win against England for the birth of his first child.
But Stokes announced his adieu on the fourth day of the match itself, shocking everyone around. But McCullum said he "tried to talk him down."
Ben Stokes shocked the world of cricket on Sunday by announcing in the middle of the third test against New Zealand in Nottingham that it was his last international.
New Zealand boosted their chances of qualifying for next year's World Test Championship final after beating England in the third and final Test in Nottingham on Monday by 160 runs, clinching the series 2-1.