Centre-right leader Luis Montenegro has been appointed as Prime Minister of Portugal but was reported to form a coalition government and grapple with growing far right, after his party's narrow victory in parliamentary elections.

The 51-year-old lawyer and veteran parliamentarian succeeded the Socialist Party's Antonio Costa, who was in power since 2015.

His Democratic Alliance had campaigned on promises to boost economic growth by cutting taxes, and to improve unreliable public health services and education, which have been hit by strikes by teachers and school workers over pay.

The newly elected leader told reporters after a meeting with President Marcelo Rebelo de Sousa, noting that Montenegro would present his new government next Wednesday, which is due to take office on April 2.

Their meeting marked the end of the process of naming a new prime minister.

Montenegro's AD beat the incumbent Socialist Party (PS) in March 10 elections but won just 79 seats, far short of a majority in the 230-seat parliament.

Meanwhile, with the backing of the tiny business-friendly Liberal Initiative party, the AD would still need the support of anti-immigration party Chega to reach a majority of 116 seats in the assembly.