Lancashire Police said they received reports over the weekend of messages sent to members of Wyre Council.
A 49-year-old man from Manchester and a 20-year-old woman from Oldham have been arrested on suspicion of malicious communication offences.
The man has been released on bail pending further investigation until May 12, while the woman remains in custody.
On Monday, Wyre Council removed the contact details of councilors from its website due to “inappropriate emails and phone calls” about the disappearance of Ms Bulley.
It said it had temporarily removed contact details for “members of the parish and town council” after Lancashire police confirmed their investigation.
Wyre Council leader Michael Vincent said: “We appreciate the emotional gravity of the situation; however, we will not tolerate any form of abuse of any of our elected members of Wyre Council or any of the town and parish councils within our borders or of our staff.
“The community has shown great strength and resilience during the investigation, and we urge everyone to continue to show compassion and empathy.
“It is unfortunate that we have had to take this step at such a difficult time and appropriate steps are being taken to ensure residents are still able to communicate with their elected representatives.”
The council added: “Any reports of abuse will be recorded by Wyre Council and forwarded to the police. Please be respectful.”
Chief Executive Garry Payne said: “Everyone is deeply saddened by the ongoing case of Nicola Bulley and our thoughts are with family and friends.
“We support Lancashire Police where needed and urge everyone to be respectful at this time.”
A spokesman for the force said: “We received reports over the weekend of a series of malicious messages that had been sent to members of the Wyre Council.
“Investigations led to the arrest of a 49-year-old man from Manchester, on suspicion of malicious communication offences. He has since been released on bail, pending further investigation, until May 12.
“A 20-year-old woman from Oldham has also been arrested on suspicion of malicious communication offenses and remains in police custody. Investigations are ongoing.”
Ms Bulley, 45, disappeared on January 27 in the village of St Michael’s on Wyre while walking her springer spaniel Willow after dropping off her two daughters, aged six and nine, at school.
She has not been seen since.
Police believe the mortgage adviser fell into the River Wyre within a “10-minute period” between 9:10 a.m. and 9:20 a.m. that day.
The search continues and on Thursday detectives extended it out to sea after saying finding it there “becomes more of a possibility.”