MUMBAI, July 3 (Reuters) - The Indian rupee climbed to a near two-month high versus the U.S. currency on Monday on the back of dollar offers by foreign banks, traders said.

The rupee was at 81.8175 to the dollar by 11:16 IST, up from 82.0375 in the previous session. The local currency reached 81.76, the highest since May 8.

There has been "massive (dollar) selling" by foreign banks since open, a proprietary trader at a private sector bank said.

There was talk of debt inflow related to a large conglomerate, and then "there are the usual equity inflows," the trader added.

The USD/INR pair is seeing dollar buying interest at 81.75-81.80 despite the inflows, prompting speculation that the Reserve Bank of India may have stepped in.

Price action suggested that RBI may be there, another trader said.

"But then... there will anyway be decent buying interest at these levels from importers."

The rupee received help from the move higher on the Asian peers and equities. The dollar index inched lower while the offshore Chinese yuan rose to 7.2524 following the slightly better-than-expected reading on the Caixin manufacturing PMI.

The focus this week is on the slew of important U.S. data releases. The ISM U.S. manufacturing data is due later in the day, followed by their services print on Wednesday.

U.S. private payrolls, jobless claims and the non-farm payrolls over Wednesday through Friday will provide cues on whether the labour market continues to be resilient.

Data on Friday indicated that underlying price pressures in the United States remained sticky. Excluding the volatile food and energy components, the PCE price index climbed 0.3%, matching expectations.

The rupee forward premiums inched up despite the higher U.S. yields with the 1-year implied yield at 1.71%. (Reporting by Nimesh Vora; Editing by Dhanya Ann Thoppil)