Delimkhanov, a member of Russia's lower house, is considered a close friend and part of the inner circle of Ramzan Kadyrov, the president of Chechnya, who enjoys Moscow's backing.
Kadyrov has dismissed through a spokesman any suggestion that Yamadayev's killing was linked to him.
Tamim said Chechen authorities have not co-operated with the investigation, and that "Russia is also responsible for untying the knot of this crime".
"The crime ... is 100 per cent of Chechen making and it's an operation of settling accounts (among Chechens)," he said.
|Dubai police say they will seek Interpol's help in arresting three suspects on the run [AFP]
The attack was carried out with a Russian-made gold-coloured handgun, Dubai police said, showing the media a picture of a weapon and a pair of black gloves.
The police said they were holding two suspects in connection with the killing and would seek an international arrest warrant for four others, including Delimkhanov.
The suspects, Mahdi Lournia, an Iranian, and Makhsud-Jan, a Tajik, were being held for questioning while the other suspects had fled to Russia, Tamim said.
None has been charged.
"It's very clear to us that the assassination of Sulim Yamadayev is a purely Chechen operation, which indicates settling scores in the UAE," Tamim said.
Delimkhanov told Russian news agencies that he would co-operate but said the investigation was flawed.
"The announcement of the Dubai police chief is a provocation and is aimed at destabilising Chechen society," he was quoted by RIA-Novosti as saying.
"The police were unable to conduct a quality investigation. ... I am prepared to co-operate with the investigation into this crime and will answer all concrete questions. However, I will demand accountability for this clear slander as the law dictates."
A source at the public prosecutor's office in Moscow said that whatever the outcome of the investigation, Russia would not extradite any Russian nationals should there be any such request.
Russian analysts say Yamadayev's killing death removed one of the last remaining powerful opponents of Kadyrov's increasingly strong control over Chechnya.
A former Chechen separatist, Yamadayev switched sides in the late 1990s and became the commander of Vostok, an elite battalion with reputed links to Russia's powerful military intelligence agency, which fought the rebels.
|Kadyrov has dismissed any suggestion that the Dubai killing is linked to him [GALLO/GETTY]
He was honoured with Russia's highest decoration, the Hero of Russia award.
Yamadayev was dismissed from the military late last year amid rivalry with Kadyrov, and Russian police issued an arrest warrant against him over the kidnapping of a Chechen businessman in 1998.
Yamadayev and his family left Russia after his brother Ruslan was shot and killed during a busy afternoon rush hour in September just steps away from Russia's main government building in Moscow.
Kadyrov worked as the head of his father's security force, which was accused of kidnapping, sadistic torture and murder.
After his father was killed by a bomb in 2004, power passed to Kadyrov.
String of killings
Vladimir Putin, then Russian president and now prime minister, has embraced the younger Kadyrov.
But Yamadayev's killing in Dubai is the latest in a string of assassinations targeting Kadyrov's opponents in and outside Russia.
Some have been shot dead on the streets of Moscow, including Anna Politkovskaya, a famous Russian journalist, whose death in 2006 shocked the world.
In January, a former bodyguard of Kadyrov was shot dead in Vienna, the Austrian capital, after filing a criminal complaint against him in June 2008, accusing him of torture.
Kadyrov has denied any involvement in the killings.