"The Ayodhya dispute is with the Supreme Court and there is a longstanding judicial directive to maintain status quo at the site where the Babri Masjid stood," said an editorial in the CPI-M's People's Democracy.
It said the Uttar Pradesh chief minister had instructed the district authorities to be vigilant.
"This is not enough. The Uttar Pradesh government should direct the district administration to stop any activity such as importing stones into Ayodhya which is meant to disturb the status quo. The mischief planned must be nipped in the bud."
Last week, two trucks of sandstone from Rajasthan arrived in Ayodhya as part of the stockpile of stones meant for the construction of the temple.
The Vishwa Hindu Parishad-run workshop had been engaged in preparing the stones for the temple construction.
VHP supporters razed the 16th-century Babri Masjid in Ayodhya in 1992, saying it had been built at the site where Lord Ram was born. They then quickly built a makeshift Ram temple at the spot.