Tuesday's meeting marked Gul's first visit to Washington as president.
Before the meeting, Dana Perino, White House spokeswoman, said Bush would encourage Gul to work with Iraqi government leaders on a long-term political solution.
Bridge across cultures
Bush said: "I strongly believe that Europe will benefit when Turkey is a member of the European Union. I view Turkey as a bridge between Europe and the Islamic world, a constructive bridge.
"And so I believe it's in the interests of peace that Turkey be admitted into the EU."
Bush reiterated that Washington would continue to help Turkey fight their "common enemy", the Kurdistan Workers Party (PKK), but that the White House had also urged Ankara to find a political solution to the separatist issue.
Turkey, which has been waging an aerial bombing campaign against PKK positions in northern Iraq, blames the fighters for the deaths of nearly 40,000 people since 1984 when the PKK took up arms to fight for a separate homeland in southeastern Turkey.
Bush said: "A common enemy is the PKK. It's an enemy to Turkey, it's an enemy to Iraq and it's an enemy to people who want to live in peace.
"The United States, along with Turkey, are confronting these folks and we will continue to confront them."
Gul said: "We are working against our common enemy, the PKK. And we have once again underlined the importance of our cooperation in fighting against the PKK."
White House officials said Turkey had shown restraint in its military response to attacks by the PKK and called on Ankara to seek open dialogue with Iraq to resolve problems along the two countries' border.
The US and the EU, like Turkey, classify the PKK as a terrorist organisation.
Turkey has as many as 100,000 troops along its mountainous border with Iraq and has recently carried out military operations with assistance from US intelligence.
Meanwhile, the death toll from a bomb attack last week in southeast Turkey's largest city rose to six on Tuesday, while Turkish authorities have detained seven people in connection with the explosion.
The PKK said on Monday that its own members working independently may have been responsible for the bomb, the first semblance of a claim of responsibility since the attack.