U.S. Brig. Gen. Vincent Brooks singled out Tikrit on Tuesday as one of the last Iraqi "strongholds."
Saddam, if he is still alive, may try to flee to his birthplace in the hope tribesmen with blood ties to him will fight to the death for their leader. Even if he is dead, experts fear Tikrit would remain a hotbed of resistance long after a new government is installed.
The town owes a lot to Saddam. Before his Baath Party came to power in 1968, it was a backwater. Thanks to government investment in infrastructure and business — largesse straight from the man at the top — Tikrit has grown into a sprawling town of 260,000 people.
Tikrit now hosts a huge army garrison for the Republican Guard, Iraq's best-trained troops, as well as an air base and air force academy.
Saddam has also studded the town with some of his largest and most elaborate presidential compounds. If he went into hiding there, regime watchers say he could easily vanish into the labyrinth of underground tunnels believed to link those sites to the eastern banks of the Tigris River.
Brooks said neutralizing Tikrit is an important step in breaking Saddam's power base. Although he would not go into details, Brooks said coalition forces were already hitting "command and control" facilities in the city.
The crash Monday near Tikrit of an American F-15E fighter jet was one sign of how the focus of the conflict may be shifting to Tikrit. If shot down, it would be only the second coalition plane taken out by Iraqi fire.
Central Command said the cause of the incident was under investigation and rescue crews were searching Wednesday for the crew.
U.S. special operations forces have been watching the roads leading north out of Baghdad to Tikrit to prevent possible attempts by Saddam to flee or any effort by the Republican Guard to regroup there.
But the Patriotic Union of Kurdistan, one of two main Iraqi Kurdish groups opposing Saddam, claimed on Tuesday that Saddam already was hiding in Tikrit. The party's newspaper said Saddam, his two sons and many of his top aides moved there after fighting intensified in Baghdad.
How much resistance Tikrit could muster is not clear.
Brooks said Tuesday that many of the Iraqi forces stationed in Tikrit early in the war have since moved south to meet the allied advance. And coalition troops have so far been able to overwhelm much bigger cities like Baghdad and Basra.
On the other hand, Tikrit is a power center for Sunni Arab tribes that may hold out for as long as possible out of fear of losing power to the nation's Shiite majority.
CBS News National Security Correspondent David Martin reports Tikrit is also considered a possible site for Iraq's alleged stockpiles of chemical or biological weapons, meaning that if Saddam flees there, there will still be the long-held fear that he might deploy such weapons as a last resort.
And if Saddam goes into hiding, he may be able to organize clandestine cells and launch a guerrilla war against U.S. troops similar to the one his Baath party waged in the 1950s.
"It is entirely possible that some residual elements of the Baath Party will reconstitute themselves as an underground revolutionary armed-struggle party that will launch terrorist attacks against U.S. forces and the interim authority," said John Pike, a military analyst at GlobalSecurity.org.
"Pacifying Tikrit is going to be a particular problem," he said.
In the Baath Party's early days, Saddam rose to prominence in 1959 by participating in a failed assassination attempt on then-Prime Minister Gen. Abdel-Karim Kassem.
Even back then, Tikrit served him well. After the botched assassination attempt, the young Saddam fled his hometown on horseback across the desert for neighboring Syria.