The reassurances came in telephone conversations initiated by Sisi, after a satellite television station aired the audio recording in which he apparently suggests that Gulf monarchies have more money than they need and that Egypt should have some.
The official Saudi Press Agency said on Monday that King Salman assured Sisi that Riyadh's support for Cairo was "unchangeable" and that nothing could harm the "strategic and exemplary" links between the two sides.
Abu Dhabi Crown Prince Mohammed bin Zayed al-Nahayan told Sisi that "any attempts of hatred will not impact on relations between the two countries, which are strong and growing".
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Quoted by the official WAM news agency, Sheikh Mohammed said the United Arab Emirates stood alongside Egypt in its efforts to "strengthen security and its development".
Sisi has already had similar telephone conversations with the rulers of Bahrain and Kuwait.
Saudi Arabia, the UAE and Kuwait are the main financial backers of Sisi's government, having pledged around $12 billion to it since he came to power.
Diplomatic relations between Egypt and Qatar have thawed since the Egyptian army deposed Islamist president Mohamed Morsi in 2013 and launched a crackdown on his Doha-backed Muslim Brotherhood movement.
The dispute triggered a crisis between Qatar and its Gulf neighbours of Bahrain, Saudi Arabia and the UAE in March. It lasted until November, when the three nations reinstated their envoys to Doha.