Alongside regional leaders, US Secretary of State Antony Blinken will attend the meetings
The first two matches, in Ahmedabad and Hyderabad, ended in draws. Brief Scores: New Zealand: 193 and 175; India 566-8 decl.
India were on the brink of claiming a series-clinching win after some inspired spin bowling reduced New Zealand to 140-8 at lunch on the fourth day of the third and final Test on Tuesday.
Spinners Pragyan Ojha, Harbhajan Singh and Suresh Raina shared six scalps while paceman Ishant Sharma picked one in a dramatic New Zealand collapse at the Vidarbha Cricket Association Stadium.
Tailenders Tim Southee (13) and Andy McKay (three) were the not out batsmen at the break, with the visitors still needing 233 runs more to avoid an innings defeat.
New Zealand suffered early setbacks after resuming the day on 24-1.
Ojha (2-45) picked off his first victim in the second over of the morning, trapping overnight batsman Brendon McCullum (25) leg before wicket with a ball that pitched on the middle stump.
McCullum, who came to open the second innings after batting at number eight in the first due to back spasms, finished as the highest run-getter for New Zealand with a tally of 370 from the three Tests.
Ojha then claimed Martin Guptill for a golden duck in his next over to leave New Zealand tottering at 38-3.
The bowler, however, missed out on a third wicket after his appeal for a leg before off Ross Taylor was turned down by umpire Simon Taufel. TV replays showed the ball would have hit the middle stump.
Harbhajan (3-43) joined in the action soon, removing struggling wicket-keeper batsman Gareth Hopkins (eight) who tried to drive the ball away but ended up being caught brilliantly by a diving Gautam Gambhir at short leg.
Taylor (29) was caught by substitute Cheteshwar Pujara at the same position off Harbhajan although replays showed the ball had bounced off his pads.
Rookie middle-order batsman Kane Williamson (eight) was bowled by Sharma while part-timer Raina had key batsman Jesse Ryder caught at mid-off for 22.
Ryder, battling a calf injury, faced 53 balls and hit one boundary during his nearly three-hour stay at the crease.
Raina's off-spin fetched him his second wicket, that of Kiwi captain Daniel Vettori who was hit on the knee roll in front of the middle stump.
The visitors added 116 runs in the session at the cost of seven wickets.
The series is tied 0-0 after the first two matches ended in draws.
Alongside regional leaders, US Secretary of State Antony Blinken will attend the meetings
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