Director Theresa Fyffe said many health visitors who carry out the screening reviews were suffering from burnout because of overwork.
She warned that the service could be forced to prioritise which youngsters are seen.
Ms Fyffe spoke out three months after the 27 to 30 month tests were reintroduced by health boards with no additional resources from the Scottish Government to carry them out.
She said: "We have concerns about the capacity within the workforce to implement these checks.
"While some boards have looked at bringing in support staff to ease the pressure on health visitors, we know from speaking to staff on the ground they are under a great deal of pressure and many health visitors are experiencing burnout."
"The last thing we want is for health visitors and boards to have to start prioritising the work they are able to do, putting the universal health visiting service at risk."
She would not be drawn on how health visitors would choose which children would be seen, but previously experts warned children from better off families had slipped through the net when the tests were not universal.
The checks assess children's language development, their sleeping and eating habits as well as other skills such as potty training.
Prior to April only children from poorer backgrounds had routine reviews from health visitors after their first birthday.
The screening for older pre-school children was made universal in April following research by the Royal College of Speech and Language Therapists which found many toddlers with language problems were not being identified.
It warned issues were not being picked up until they started school, at which stage they were harder to correct.
Kim Hartley, Scotland Officer for the Royal College of Speech and Language Therapists, said it was important all children should be seen by a health visitor at 27 to 30 months to ensure language problems were being picked up.
She said: "Speech, language and communication difficulties are the most common difficulty children and young people face growing up regardless of their home circumstances.
"We know without universal screening children with difficulties can fall through the net compromising not just their access to education but their mental health, enjoyment of rights, social connectedness and ultimately life outcomes.
"If Scotland is to become the best place to grow up we need to be the best at optimising every child's speech, language and communication development from pre-birth and detecting and managing difficulties as early as possible when they appear."
Jackson Carlaw, Scottish Conservative health spokesman, said: "The Scottish Government pays lip service to the importance of early years and health visiting, but it needs to match these words with action.
"It is unacceptable that they are not given full backing to do this.
"I urge the Scottish Government to address these problems at once."
Tam Baillie, Scotland Children's Commissioner, has previously criticised ministers for failing to recruit enough health visitors to carry out the checks.
The 27 to 30 month reviews were scrapped several years ago, but reintroduced in April in a policy change.
A Scottish Government spokesman said: "NHS Boards started 27-30 month health reviews in April in line with our guidance, and we expect NHS Boards to plan their workforce to deliver these reviews.
"Boards should bear in mind the work of the Early Years Taskforce, which encourages partners to bring the totality of their resources to discussions on how best to support preventive and early intervention work."