"To start presidential and parliamentary elections as soon as possible, is what will bring the country out of crisis," Saleh's letter obtained by Sputnik news agency said.
The ex-president reassured Hadi that he and his son do not consider the acting president to be their enemy.
Saleh also asked Hadi to cancel "emotional decisions" to freeze the accounts of the General People's Congress, headed by the former president, made by the UN Security Council, the United States and personally by the acting president in November 2014.
Fierce fighting continued on Wednesday in and around Sanaa after Houthi rebels stormed the presidential residence overnight.
The Houthi are a major opposition force in the country and were instrumental in removing former Yemeni leader Ali Abdullah Saleh from office in 2012. In September 2014, the insurgency spread to Sanaa and the central and western parts of the country.
