US telecoms company MCI has accepted an improved $7.6bn (£4bn) cash and shares takeover offer from rival Verizon.
Both Verizon and Qwest have increased their initial offers for MCI
Verizon is trying to end a bidding war with rival suitor Qwest Communications, which has offered $8.45bn.
MCI's management continues to back Verizon, in the latest round of what has become a wholesale consolidation of the US telecoms business.
But Qwest could still make a hostile bid direct to shareholders, some of whom are keen for the firm to win.
"Qwest has a superior offer," said key MCI shareholder Bruce Berkowitz. "The highest bid should win."
Verizon and Qwest have been competing to buy MCI since the start of February.
Qwest has given MCI until 5 April to accept or reject its offer.
It said on Tuesday that its bid remained the better of the two, and that it would "assess the situation".
Verizon's chairman and chief executive Ivan Seidenberg, said that a combined Verizon and MCI would "create a formidable and highly competitive company".
Both Verizon and Qwest have now increased their initial cash and shares offers for MCI, which is the second-biggest long distance phone firm in the US.
Verizon's initial offer totalled $6.75bn, while against $8bn from Qwest.
MCI had earlier accepted the first Verizon offer on 14 February, but a revolt by MCI shareholders scuppered the deal after they launched a legal action, claiming that it undervalued the company.
Mr Berkowitz, whose Fairholme Fund owns 11 million MCI shares, said on Tuesday that Verizon's newly raised offer was "a step in the right direction".
He described the ongoing takeover saga as "not over".
"MCI is hugely valuable to both Qwest and Verizon," he added.
MCI was formed in 2002 from the rubble of WorldCom, after it filed for the largest bankruptcy in history following an $11bn accounting fraud.
It has its roots in long-distance services, and is regarded as valuable for its one million business customers, and its large network which also carries much of the world's internet traffic.
Phone company Verizon is a powerful player in north-east United States, and has a joint mobile venture with the UK's Vodafone. A deal with MCI would give it a wider national presence.
Qwest is the local phone carrier in the less-densely populated Rocky Mountains and north-west of the US.