It also expressed scepticism over Tillerson's assurances regarding putting pressure on Pakistan to act against the terrorist groups operating from the country.
Former CPI(M) general secretary Prakash Karat, in an editorial in the forthcoming issue of party organ 'People's Democracy", called the visit "another step in the US gambit to draw India fully into a strategic-military alliance in what is called the Indo-Pacific region".
Mr Karat pointed out that before his visit to India, Tillerson in his speech at a think-tank in Washington had said that "the United States and India are increasingly global partners with growing strategic convergence".
He said that Tillerson's speech, titled 'defining our relationship with India for the next century', had made it clear that the United States wishes to harness India in its efforts to contain China in the Asia-Pacific region.
"He accused China of 'undermining the international rules-based order' even while praising India's role. Tillerson underlined the fact that the foundation of the strategic alliance is the military relationship," Karat said.
The Left leader alleged that "Tillerson made it clear that the United States wishes to harness India in its efforts to contain China in the Asia-Pacific region....
"He accused China of 'undermining the international rules-based order' even while praising India's role."
On Tillerson's assurances regarding putting pressure on Pakistan to act against terror groups, Mr Karat reminded that the top US official arrived in Delhi after visiting Islamabad and that the Trump administration has always acknowledged Pakistan's importance in resolving the conflict in Afghanistan.
"This was done, as usual, to allay India's concerns, but the reality is that the Trump administration cannot dispense with the Afghanistan and Pakistan strategy," the Left leader wrote.
He cautioned that Tillerson's visit was also aimed to get India on board for its sanctions and hostile manoeuvres against Iran as Trump administration "is determined to scuttle the nuclear agreement arrived at with Iran by the Obama administration".
"Going along with the US sanctions on Iran would be harmful to India's vital interests in the energy and economic spheres, but given past experience, the Modi government may give in to US pressure," Mr Karat observed in the editorial.
Mr Karat has cited that "earlier dialogue between the foreign and commerce ministers of both the countries are now replaced by dialogue involving the foreign and defence ministers highlighting the military strategic nature of the relationship".
Along with selling more weaponry worth billions of dollars to India, US administration has also put pressure to sign other foundational military agreements like the CISMOA, he claimed.
Tillerson arrived in India on October 24 on his maiden visit. During his three day stay, he held talks with the Indian leadership on further strengthening the Indo-US strategic partnership.