The hurricane made landfall in the state a week ago, causing devastating floods.
Some residents have been allowed to return to their homes but flood waters are still rising in other areas.
Harvey has been blamed for at least 47 deaths, and about 43,000 people are currently housed in shelters. Mr Trump has asked Congress for $7.8bn (£6bn).
The sum would be an initial payment to help with recovery efforts following the flooding in both Texas and Louisiana, which has also hit production at America's main petrol and oil refining centre.
Governor of Texas Greg Abbott has said the state may need more than $125bn in aid.
President Trump and his wife Melania visited Texas earlier in the week but stayed clear of the disaster zone, saying they did not want to divert resources from rescue efforts.
However, the president was criticised for not meeting victims of the flooding and for focusing largely on the logistics of the government response.
President Trump has declared Sunday a "National Day of Prayer" for victims of Hurricane Harvey.
Emergency funds request
Administration officials say there will be further requests for funds when the full impact of Hurricane Harvey becomes known.
In a letter to House of Representatives Speaker Paul Ryan, White House budget director Mick Mulvaney warned that failure to raise the US debt ceiling could hinder recovery efforts.
The debt ceiling is a cap on the total amount the US government can borrow. Only Congress can raise that limit.
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