Rajput, 45, played two Tests as an opening batsman in 1985 and served Mumbai with distinction in first-class cricket before being given charge of the Indian under-19 team last year.
Rajput replaces Chandu Borde, the manager for the ongoing Test and one-day tour of England, as India continue stop-gap arrangements in the absence of a regular coach since Greg Chappell resigned in April.
Former captain-turned-commentator Ravi Shastri managed the team on the tour of Bangladesh in May but declined to continue, citing media commitments.
The Board of Control for Cricket in India (BCCI) said on Wednesday it will advertise for the post of national coach within the next week.
"We want to get a coach as early as possible," BCCI spokesman Rajiv Shukla told reporters after a meeting of its top officials here.
Wicket-keeper Mahendra Singh Dhoni was on Tuesday appointed captain of a revamped Indian team for the Twenty20 world meet in Johannesburg, Cape Town and Durban from September 11-24.
Regular captain Rahul Dravid, Sachin Tendulkar, Sourav Ganguly and Zaheer Khan opted out of the tournament to rest ahead of a busy season that features matches against Australia and Pakistan.
India are drawn with Pakistan and Scotland in group D of the preliminary league of the Twenty20 competition, with the top two advancing to the Super Eights round.
The tournament features the nine nations currently playing Test cricket along with Zimbabwe, Kenya and Scotland.