US officials are looking for Ahmad Khan Rahami, a naturalised US citizen who was born in Afghanistan, for questioning over explosions in New York and New Jersey on Saturday.
The FBI warned that the suspect is "armed and dangerous".
Further devices exploded in Elizabeth, New Jersey, in the early hours of Monday as a police robot investigated a bag. No-one was injured.
The FBI has been searching the 28-year-old suspect's family home in Elizabeth.
The bomb that hit Manhattan's Chelsea district on Saturday evening injured 29, and an unexploded device was found nearby.
Early on Saturday a pipe bomb exploded in a New Jersey shore town ahead of a charity race. No-one was hurt.
Democratic presidential candidate Hillary Clinton has urged jittery Americans to be "vigilant but not afraid".
"It is crucial that we continue to build up trust between law enforcement and Muslim American communities," she said.
New York Governor Andrew Cuomo said there were "certain commonalities" between the bombs.
New Jersey State Police said that the FBI was linking the Manhattan explosion to that in Seaside Park, New Jersey.
Millions of New York residents received a phone alert on Monday morning after the Notify NYC alert system was used to name the suspect.
"Anyone who sees this individual or knows anything about him or his whereabouts needs to call it in right away," New York Mayor Bill de Blasio said on CNN.
Both the bomb that went off in New York on Saturday and the unexploded device found four blocks away were shrapnel-filled pressure cookers, according to US media.
Mr Cuomo said on Monday that there might be a foreign connection to the Manhattan attack. Officials had said over the weekend there were no confirmed links to international terrorist groups.
President Barack Obama, who is in New York for the United Nations General Assembly, is being kept up to date with the investigation, a White House spokesman said.
Early on Monday, a New Jersey police robot inspected a backpack found in a bin near the railway station in Elizabeth, causing it to explode.
The bag had been picked out of the bin by two men on Sunday evening. They thought it could contain something of value but saw wires and notified the police.
"That was not a controlled explosion," said Elizabeth Mayor Christian Bollwage.
These are the attacks in the US over the weekend that caused security alerts:
- A pipe bomb exploded on the route of a charity race in New Jersey on Saturday, forcing the event to be cancelled but causing no injuries
- Also on Saturday, nine people were stabbed at a shopping mall in the US state of Minnesota, in an attack apparently carried out by a 22-year-old ethnic Somali.
- At least 29 people were injured in an "intentional" explosion in the Chelsea area of Manhattan at around 21:00 (01:00 GMT on Sunday)
- Another device was later found four blocks from the site of the blast and destroyed in a controlled explosion
- A backpack containing suspicious devices was found in Elizabeth, New Jersey, late on Sunday
- One device found in Elizabeth exploded early on Monday as a bomb disposal robot tried to deactivate it