Letta was due to give a press conference later on Wednesday to address the rumours of a possible "relay" in the centre-left Democratic Party (PD), which elected the now 39-year-old Renzi to be its leader in December.
"Letta-Renzi: The Final Duel" read a headline in La Stampa daily, while the top-selling Corriere della Sera said: "Letta holds out but Renzi government looms".
Renzi has been critical of Letta, who only came to power in April at the head of a coalition with the right and has struggled to drag Italy out of a painful recession that has pushed unemployment to record highs.
Renzi has accused Letta of being too slow in taking decisions and lacking the courage to confront key problems, such as the cost of Italy's bureaucracy.
Letta meanwhile has reportedly accused Renzi of "betrayal" since there had been a tacit deal that Letta would stay in office until at least the end of 2014.
A majority of Italians would be against the Letta-Renzi switch - 68 percent according to one recent poll.
The central leadership of the Democratic Party is due to meet on Thursday - a meeting that could prove decisive on embattled Letta's fate.