He extended the offer in a phone call on Friday with the Palestinian Authority president, their first contact since Mr Trump took office.
Mr Trump hosted Israeli Prime Minister Benjamin Netanyahu in February.
At that meeting, Mr Trump dropped a long-standing US commitment to a two-state solution for the Israeli-Palestinian conflict.
What is the two-state solution?
A “two-state solution” to the decades-old conflict between Israelis and Palestinians is the declared goal of their leaders and the international community.
It is shorthand for a final settlement that would see the creation of an independent state of Palestine within pre-1967 ceasefire lines in the West Bank, Gaza Strip and East Jerusalem, living peacefully alongside Israel.
The policy shift fuelled Palestinian concerns that Washington was becoming even more pro-Israeli under the Trump administration.
There have been no substantive peace talks between the Israelis and Palestinians since US-mediated negotiations broke down in April 2014.
Mr Trump’s spokesman, Sean Spicer, confirmed on Friday that the US president had invited Mr Abbas to the White House “very soon”.
The president told the Palestinian leader they would discuss how to rekindle peace talks, Abbas spokesman Nabil Abu Rdainah was quoted as saying by Reuters news agency.
He said Mr Trump had stressed “his commitment to a peace process that will lead to a real peace between the Palestinians and the Israelis”.
Palestinian officials had indicated before Friday’s call that Mr Abbas would press the US president about Israeli settlement-building on occupied land and the importance of a two-state solution.
During last month’s joint press conference, Mr Trump asked Mr Netanyahu to “hold back on settlements for a little bit”, appearing to catch the Israeli leader unawares.
The US Senate on Thursday approved Mr Trump’s controversial nominee for ambassador to Israel.