Banca Monte dei Paschi di Siena SpA, the world’s oldest bank and a festering wound of Italy’s financial system, may finally be heading for a sale.

Italian bank UniCredit SpA said Thursday it is entering into exclusive talks with Rome on a possible takeover of the storied Monte dei Paschi, whose history stretches back to its founding in 1472 during the days of the Italian Renaissance.

Once the third-largest bank in Italy, Monte dei Paschi became the poster child for the problems in European banking last decade. It put itself up for sale in 2014 but failed to find a suitor.

After the bank teetered near failure for years, Rome spent €5.4 billion to nationalize it in 2017, equivalent to more than $6 billion at the time. Monte dei Paschi has long been weighed down by mountains of bad loans and a legal scandal. Italy promised European regulators to return Monte dei Paschi to private hands by April 2022.

With the deadline looming, the government has been pushing UniCredit, Italy’s second-largest bank, to do a deal.

A person familiar with the matter said the treasury’s talks with UniCredit and other potential buyers had recently intensified and now a deal could happen in the next few weeks. The price tag could be as low as a symbolic €1, similar to previous deals orchestrated by the government in which ailing Italian lenders were sold to larger rivals.

Italy’s government been pushing UniCredit, Italy’s second-largest bank, to do a deal.

Italy’s government been pushing UniCredit, Italy’s second-largest bank, to do a deal.

Photo: Antonio Pisacreta/Zuma Press

Andrea Orcel, a mergers and acquisitions banking veteran from UBS Group AG, became UniCredit’s chief executive earlier this year. UniCredit said it agreed to negotiate about selected parts of the Monte dei Paschi, but agreed with the Italian treasury it would only pursue a deal that had no impact on its capital position.

Monte dei Paschi’s existing bad loans would also need to be excluded from the acquisition. UniCredit would also be shielded from the multibillion legal costs stemming from the Tuscan bank’s numerous ongoing litigations.

The takeover of Monte dei Paschi by a larger and more solid bank would put an end to years of limbo for one of Europe’s most problematic lenders. It failed repeatedly to sell itself last decade and burned through billions of euros in investor capital. The value of its shares all but evaporated.

The European banking industry is highly fragmented, with each country hosting many regional banks. Italy’s economy has barely grown for years, creating a difficult environment for banks to operate. The possibility of rising soured loans resulting from the damage inflicted by the Covid-19 pandemic looms on the horizon.

Regulators have long called for Italian banks to merge to strengthen themselves. Consolidation is seen as a way for banks to reduce costs and slash their way to more profits.

Italy’s government led by Mario Draghi, a former European Central Bank’s president, recently simplified procedures for banks to benefit from tax breaks when they merge, effectively giving them more time to access these benefits.

Mr. Orcel, who took the helm of UniCredit in April, has been focusing on simplifying the bank’s managerial structure, expressing little interest in acquiring other banks.

He said he would decide whether to make a deal or not likely in September, after due diligence of a few weeks.

UniCredit said on Friday that net profit for the second quarter surged to €1.03 billion, equivalent to $1.22 billion, from €420 million a year earlier, helped by higher revenue and lower provisions for bad loans.

Write to Giovanni Legorano at giovanni.legorano@wsj.com