BANGKOK, June 27 (Reuters) - Thailand's leading prime ministerial candidate Pita Limjaroenrat said on Tuesday he has enough support in the upper house to become the country's next premier, just days ahead of the new parliament's first session.

Pita, the leader of the progressive Move Forward Party, faces an uncertain path to the premiership despite scoring a stunning victory in a May poll that saw Thais reject nearly nine years of military-backed government.

His eight-party alliance together has 312 seats in parliament. Under the constitution, to become prime minister, Pita needs at least 376 votes in a joint sitting of the bicameral legislature, including the 250-member upper house, most of whom were chosen by the military when it took power in 2014.

When asked on Tuesday how much Senate support he had secured, Pita said: "enough for me to become prime minister".

Doubts have lingered over whether Pita has enough support because of his party's controversial proposal to amend Thailand's strict royal insult law or lese majeste. Move Forward has said the law, which prescribes up to 15 years of jail for perceived offences against the monarchy, is used as a political tool against opponents of the current government.

The stance has antagonised country's royalist establishment and old-money elite, including the conservative-leaning Senate.

The party was in the process of explaining its position to senators ahead of the July parliamentary vote, said Pita.

"Amending the law in keeping with society's context is not something that will stop government formation," he said.

After convening on July 3, parliament is expected to vote on a prime minister on July 13.