TUCSON, Arizona, (AFP) - US President George W Bush and Iraqi Prime Minister Nuri al-Maliki have agreed on "a general time horizon," not an "arbitrary timetable" for a drawdown of US forces, the White House said Friday. In talks by video-conference Thursday, Bush and Maliki said improving security conditions in Iraq should allow for including that goal in a pending long-term diplomatic and security pact, said spokeswoman Dana Perino. "Improving conditions should allow for the agreements now under negotiation to include a general time horizon for meeting aspirational goals - such as the resumption of Iraqi security control in their cities and provinces and the further reduction of US combat forces from Iraq," she said in a statement. "The president and prime minister agreed that the goals would be based on continued improving conditions on the ground and not an arbitrary date for withdrawal," she said. Earlier, an Iraqi government spokesman said Bush and have discussed a time-frame to transfer security from US forces to Iraqi troops and so be able to cut the number of American forces. The two leaders discussed the growing abilities of Iraqi troops in managing security in the country, Ali al-Dabbagh said in a statement. "They focused on having a time-frame for the complete transfer of security responsibility into the hands of the Iraqi forces in order to decrease the number of American forces in Iraq and later withdraw them," Dabbagh said. He said the two leaders vowed to work on "increasing the ability of Iraqi forces to improve the security situation on the ground so as to achieve this goal (US withdrawal from Iraq)." Dabbagh said the two leaders discussed the future of relations between Baghdad and Washington. "They talked about establishing new relations between Iraq and the US which respect the sovereignty of Iraq, and discussed the joint interests of the two peoples in areas such as health, science, culture and security fields," Dabbagh said. A US embassy official in Baghdad said the originally envisaged security pact called Status of Forces Agreement (SOFA) has now been "suspended". "The SOFA as we had in Japan or Germany has been suspended or put aside but not thrown away," he told AFP on condition of anonymity. He said the two sides are currently negotiating a "security protocol or operational protocol" that contains the key contents of the SOFA but would be for a "certain period of time." "That does not mean we would not find a more detailed, more general" agreement later, he said. The US embassy official said it was still possible that some kind of a deal would be finalised by end of the month. Bush and Maliki, meanwhile, welcomed the withdrawal this month of the last of the "surge" brigade, Dabbagh said.