India-based Apollo said it may have to bear "significant and unanticipated costs" that were "well beyond" those it was obligated to bear under the initial merger agreement.
"Cooper has acknowledged to Apollo that some price reduction is warranted. The issue now is by how much," Apollo said in a statement late on Sunday.
Cooper, however, responded saying any change in price was not warranted. On Friday, Cooper said it had filed a complaint in a U.S. court to push Apollo to close the acquisition in a timely manner.
"Cooper has not agreed that a reduction in share price is warranted," Cooper said in a statement issued late on Sunday.
Under the initial agreement, Cooper shareholders stand to receive $35 per Cooper share, a premium of more than 40 per cent to its price before the acquisition announcement, valuing the deal at $2.5 billion.
"The situations with the USW (United Steelworkers) and the joint venture partner and union in China are a direct result of the merger agreement, and are risks Apollo assumed under the merger agreement," Cooper said in a statement.
Apollo is negotiating a deal with the United Steelworkers, which represents Cooper workers in the United States, after a U.S arbitrator ruled Cooper cannot sell two of its factories in the country until a collective bargaining agreement is reached between Apollo and members of the plants' union.
"Apollo has indicated to the USW in discussions over the past two weeks that Apollo is willing to make material concessions to the USW, subject to arranging for additional financing or financial concessions," Apollo said.
Separately, workers at Cooper's China joint venture, Cooper Chengshan Tire Co, in eastern Shandong province, have been striking against the deal for about three months, while Cooper's local partner has filed a lawsuit seeking to dissolve the business arrangement.
Last month, the Chinese plant's union said the workers refused to make Cooper-branded tyres, and would stop putting the JV's production, operation and financial information into data systems.
"Cooper has misrepresented its management and control of this asset to Apollo and to its own shareholders," Apollo said in its statement.
"Cooper's efforts to establish control over its subsidiary's operations and to assert Cooper's rights against its JV partner, Apollo cannot be responsible for Cooper's failures to do so," it added.
The two companies declined further comment when contacted by Reuters.
Shares in Apollo gained as much as 3.8 percent on Monday on growing uncertainty about the status of the deal. Apollo shares have struggled since announcing the deal earlier this year as the Indian tyre maker plans to fund the acquisition entirely through debt.