"I am frustrated we haven't caught them," said Tribal Police Chief Leonard Butler. "We got so close, but we haven't been able to nab them."
The search was called off late Friday afternoon after heat, humidity and exhaustion had taken their toll on the officers.
The search was to resume late Saturday with officials expecting the suspects make another move under cover of darkness.
"Most of the activity we have noted has been at night," Butler said.
Police pursued the men Friday near Montezuma Creek who were thought to be fugitives from a Colorado cop-killing last month.
Police told Reporter Holliday Moore of CBS affiliate KRQE-TV Friday that the fugitives had appeared to split up as officers intensified their search.
Officials said they came within 15-20 yards of the two men at 2 a.m. Friday, but the two once again eluded their trackers.
CBS News Correspondent Jim Axelrod reports that a federal law enforcement official said that an FBI tactical unit will mobilize once the suspects are pinned down for sure.
The fugitives are Alan "Monte" Pilon, 30, of Dove Creek, Colo., and Jason McVean, 26, of Durango, Colo. Pilon and McVean, along with Robert Matthew Mason, 26, of Durango, are believed to have shot Cortez police officer Dale Claxton when he stopped them in a stolen water truck on May 29.
In the gun battle that ensued, the criminals wounded two sheriff's deputies, then disappeared into the desert canyon country.
Mason was later found dead of an apparent self-inflicted gunshot wound on June 4 in the canyon area known as Swinging Bridge, just hours after he apparently shot and wounded San Juan County sheriff's deputy Kelly Bradford. Three pipe bombs made of black powder were found with him, San Juan County Sheriff Mike Lacy said.
On Thursday night, the two men were seen crossing a clearing on the northern end of the Navajo reservation. They are now believed to be hiding in the canyons of southeastern Utah.
Butler said Friday the two men, if they are the suspects, may still have sidearms.
Authorities closed off the Swinging Bridge area along the San Juan River on Friday afternoon to track footprints.
The latest flurry of activity began late Tuesday night when officers saw what appeared to be a campfire burning along the river bank, Butler said. SWAT teams set up observation points around the area and waited until Wednesday morning to begin searching.
Sometime during the night Wednesday the two men apparently crossed the river and broke the police containment, again eluding officers until they spooked some livestock grazing in the brush.
Officers confirmed the movement in the brush was human and heard one suspect cough. Navajo Nation fficers again set up observation posts Thursday afternoon, at which point an officer saw them crossing the road.
Butler said the men were wearing dark clothing, but it was unclear if it was wet camoflauge fatigues. He said they appeared not to have packs or long rifles.
The 12-mile-long corridor from Bluff to Montezuma Creek has been the focal point of search efforts since a 9-year-old girl spotted two men trying to get into a water truck parked in the back yard of her uncle's home in Montezuma Creek on June 28.
Lacy confirmed that an investigation indicates that both Pilon and McVean had had dealings with the Four Corners Patriot Militia in the past.