What you need to know:
Mr Odinga, who has been invited to Mr Kenyatta's swearing-in, party, has vowed never to recognise him and his government.
Instead, he has launched a broad campaign for what he terms electoral justice, including street demos and economic boycott.
Some National Super Alliance (Nasa) supporters are unhappy with Raila Odinga's decision to reject plans to swear him in as the 'people's president'.
Some Nasa leaders wanted Mr Odinga to take oath of office on Tuesday, the same day President Kenyatta will be sworn in for his second term.
The swearing-in, using the results of the August 8 presidential election, which Mr Odinga claims he won, was to be done during the Nasa’s prayer rally for victims of police brutality at Jacaranda grounds in Nairobi.
But Mr Odinga turned down the plans, citing his international image and his respect for the rule of law.
And for that decision, some of his supporters, mostly from his Luo Nyanza backyard, are accusing him of “cowardice”.
In Migori town, a number of the residents congregated at various newspaper selling points early Monday morning and discussed the latest development with dismay written on their faces.
"We were ready to have him sworn in as our president. Why chicken out?” said Joel Omondi, a Nasa supporter from Migori town.
“This has demoralised our resistance agenda. I will now turn my attention away from Nasa activities."
Ms Rahab Matinde, a trader from Isibania, said she was utterly disappointed that Mr Odinga will not take oath of office.
"I was ready to hear baba has been sworn in no matter what. He is the people's president,” she said.
Other supporters took to social media to express their displeasure.
"If Raila does not want to be sworn in, let him retire from politics completely. Let him even leave this country for good. I did not know I have been supporting a coward all along," Kepha Omai posted on Facebook.
Abrasive lawyer Miguna Miguna, said to be one of Nasa leaders who wanted Mr Odinga to take the oath, also weighed in on the matter, expressing his dismay.
"I'll not be at the Kasarani Stadium to witness the beautification of an atrocity and the bedecking of despots. I'll not be at the Jacaranda Grounds to prostrate myself in surrender to fraudsters. I'll only mourn our heroes, heroines and martyrs AFTER LIBERATION!,” he tweeted.
Nyang Kolwalo, another Twitter user, said: "In leadership, obeying orders from both sides is good. NRM and Nasa supporters have always listened to Raila and obeyed his orders. It’s now time he obeys his supporters' orders".
Mr Odinga, who has been invited to Mr Kenyatta's swearing-in party, has vowed never to recognise him, his win and his government.
Instead, he has launched a broad campaign for what he terms as "electoral justice', including street demos and economic boycott.