Bereavement services staff were inspected at Willenhall Lawn Cemetery. at 6 a.m. Thursday, but found no clear evidence of recent damage or vandalism.
Additionally, the council said it had not received any direct complaints from parents or other family members.
The council believe old CCTV footage linked to an incident at the cemetery a year ago surfaced on social media this week.
In April, families besieged the council when staff at Streetly Crematorium removed toys, keepsakes and even photos of babies scanned from graves without parental permission.
Councilor Garry Perry, Deputy Leader of Resilient Communities, said: “I imagine this speculation has very understandably caused a great deal of concern to many bereaved families and I would like to reassure them that we are taking this matter very seriously.
“Our cemeteries are public places of quiet reflection and remembrance and it is important that visitors feel safe at these sites and that the graves of their loved ones are safe.”
He added: “Almost 12 months ago there was an incident at Willenhall Lawn Cemetery which was investigated and resulted in additional CCTV being installed.
“It appears that images taken at that time have surfaced again on social media this week and no doubt people who see these images will believe them to be recent.”
He added: “We are aware that this time of year could be of some significance to the person who caused the previous damage. Our cemetery staff will be increasing our monitoring of this area over the coming weeks.
“Any items left on the graves of babies in this cemetery are not removed by staff, even if they are in poor condition, as they are aware of the potential distress this can cause families. We recognize that these items may have great sentimental value. .
“If any parents or family members have any information or concerns, please contact our Bereavement Services team or, if they believe a crime has been committed, contact West Midlands Police.”
