Knut yesterday defied President Uhuru Kenyatta order to call off their three-week strike.
They instead vowed to dig in until the government honours its earlier pledge to implement the return to work formula by paying their dues in full.
Top Knut chiefs said they will not accept the government offer to settle their demands in two phases.
"The national executive council has sat down to deliberate on the offer by the government, and I am on instructions to say that we would like to see an improvement on the offer," Knut secretary general Madzo Nzili said.
But Nzili said Knut's is willing and ready to negotiate in a bid to find a lasting solution to the stalemate.
The standoff which enters its fourth week today has paralysed learning in public primary schools and a section of secondary schools.
Last week the government took top Knut officials to court for holding an illegal strike, claims that have been refuted by the union, which says that it was exercising its constitutional right to down tools over a breach of agreement.
Yesterday, the union officials clarified that they were only acting on behalf the teachers, whose demands the government has failed to meet.
President Uhuru Kenyatta's pleas to the educators to return to work have hit a wall as the latter maintain a hard line position. They insist that harmonized house and commuter allowances be paid prior to engaging in any other negotiations.
Although the constitution mandates all collective bargaining agreements to have been finalized by end of last month, the teachers have vowed to stay put until their demands are addressed in full.
They are calling for the implementation of Legal Notice Number 534 of 1997, which has never been fully addressed, despite earlier agreements which have ended similar strikes in the past.
Last week the Kenya Union of Post Primary Education Teachers signed a return to work formula and distanced itself from Knut.
"There is a conspiracy that Knut does not represent all teachers, and now some people are busy trying to relegate it," chairman William Sosion said and hit out at the rival union.
"Anybody thinking that they can enter into small unions to compel the teachers to call off the strike is very wrong," he said.
While accusing Kuppet of betraying them, the Knut officials said the actual offer that the government gave out, and which was signed by the former was sh3 billion and not sh17 billion as earlier reported.
"We are prepared to face the future years while sustaining the same spirit, regardless of tribe or political parties," Sosion said, adding that the union was determined to remain united.
Public opinion has been divided on whether the strike should continue, with a section supporting it and another condemning and claiming it was politically instigated.
Fears are strife that the standoff may delay the secondary schools' form four and primary standard eight mock exams, which are scheduled for August. The move may also delay the kickoff of end year final exams slated for October and November respectively.
Yesterday the officials rubbished the political claims, saying the union represents teachers from across the country.
They are demanding sh47 billion which has been in areas since a return to work agreement that was signed by the two parties in 1997, which in turn ended a similar standoff.
Earlier yesterday, there was hope that the strike would be called off, following speculations from a section of the media that Knut's NEC would meet and declare an end to the stalemate.
The lowest paid teachers in job group G earn a monthly commuter allowance of sh1001, while their counterparts in job group R take home sh4,410.
According to the revised rates, which were supposed to be harmonized, the lowest was to get sh4,000, while the highest was to bag sh16,000.
"A serious employer would propose to pay this money at once instead of breaking it down," Nzili said and challenged the government to act.
The secretary general has gone on record telling the government that teachers have the strike, while the state has money. He has pleaded with the two sides to swop their possessions: "Let them give us money, and they take the strike."
The union declined to respond to Uhuru's plea, saying "we cannot respond to presidential statements, as they are presidential statements, while ours are negotiations."
Nzili refuted claims that he and Sosion were reading from different scripts after falling out due to a document that either of them signed without the consent of the other.
"That is propaganda, we have never differed because of any document," he said.