(Adds Most new proposal on talks, paras 9-10)

ZAGREB, Dec 3 (Reuters) - Croatia moved closer to holding a new election on Thursday when parliament convened for the first time after an inconclusive vote on Nov. 8 and its speaker-elect turned down the job after failing to win cross-party support.

President Kolinda Grabar-Kitarovic held an unsuccessful first round of talks last week on forming a coalition cabinet as the minority party that holds the balance of power could not decide which party to back. She set the next round for Dec. 7.

The opposition conservative HDZ party won 59 seats in the 151-seat parliament, three seats more than the incumbent Social Democrats-led centre-left coalition. Reformist newcomers “Most”, Croatian for “bridge”, has 19 seats.

Whoever wins the support of at least 76 deputies will become prime-minister designate.

The Social Democrats (SDP) put forward Most candidate Robert Podolnjak for speaker but he refused the job because HDZ did not support him.

It is unclear when the parliament will convene again.

“This situation indicates that we could face new election as insistence by Most to win support of both major parties looks unrealistic. Also, Most may assess that they could fare even better in new election,” political analyst Ivan Rimac said.

Most, founded three years ago and largely made up of municipal politicians, insists on a joint government and wants to overhaul the public sector and judiciary, and reduce taxation pressure on businesses as well as fiscal imbalances.

On Thursday, Most invited both major parties to talks next Monday to discuss forming a reformist government.

“If just one side turns up, we will continue talks with that side on forming a reformist government. We act responsibly and it would be HDZ’s or SDP’s fault if we go to polls again,” Most said in a statement.

“The current political situation is not unusual for a democracy, but the problem is that Croatia needs a new cabinet to urgently tackle fiscal woes and uncompetitive economy. At the moment we don’t know the future economic policy and even don’t have a budget for 2016,” an economic analyst Damir Novotny said.

Croatia’s public debt is close to 90 percent of GDP and employment is at 17 percent. The European Commission and the International Monetary Fund want the next government to reduce debt and barriers to investment, notably in the private sector. (Editing by Louise Ireland)