BUENOS AIRES, May 27 (Reuters) - Argentina's markets have a wary eye on President Javier Milei's sprawling reform bill that is grinding through Congress and facing likely amendments in the Senate that will push it back to the lower house and gum up its final approval.

The country's bonds and peso, which rallied strongly in the early months after libertarian economist Milei took office in December, have ceded ground in recent weeks, weighed down by delays to the signature bill and rising political noise.

The reform package, ranging from plans to privatize state bodies to measures to encourage investment, is a key plank in Milei's plans to fix the economy with inflation near 300%, myriad capital controls, and depleted foreign currency reserves.

Its fate, therefore, is key for markets which have generally cheered Milei, a former TV pundit who won a shock election last year when he pledged "chainsaw" spending cuts to overturn an entrenched fiscal deficit after years of debt crises.

"The rejection of the 'bases' bill and tax bill would confirm the political weakness of the government in building a legislative majority," said economist Joaquín Marque at local financial services firm UG Valores.

"That would generate greater uncertainty in various macroeconomic conditions and postpone the economic recovery."

While Milei's austerity measures have boosted the country's fiscal position, Argentina's economy has tanked, with economic activity and construction down sharply, hit by moves including to halt public works. Poverty and dollar prices are rising fast.

Consultancy Portfolio Personal Inversiones, meanwhile, said "positive news in the Senate about both bills and a greater inflow of agriculture sales" could boost the peso in widely used parallel markets to get around capital controls.

Milei, who only has a minority in Congress, has already conceded the bill is likely to take longer to move forward than his government had hoped, which has already delayed an important pact with regional governors Milei had aimed to sign in May.

A government adviser close to Milei, asking not to be named, said that the date of the pact with governors was less important than striking an agreement that would be long-lasting.

"We don't care about the date, it could be June 20, July 9 or Oct. 17," the person said last week, adding that the reform bill should first get congressional approval. The bill likely faced modification and being sent back to the lower house, he added.

"There is no rush to sign the pact... It is preferable to wait for Congress to approve the bases law and to then advance a majority agreement between governors, legislators, union leaders and the church, among other social actors."

(Reporting by Walter Bianchi; Editing by Adam Jourdan and Andrea Ricci)