The Mass and contemplative prayer naturally fit together, and it is possible to engage in this type of prayer during the liturgy.
It's also helpful to look at the structure of the Mass for inspiration on how to engage in our own individual contemplative prayer.
The Catechism of the Catholic Church makes this connection in its section on prayer, stating, "Entering into contemplative prayer is like entering into the Eucharistic liturgy" (CCC, 2711).
The Catechism then compares the structure of the two, going through it step by step:
[W]e "gather up" the heart, recollect our whole being under the prompting of the Holy Spirit, abide in the dwelling place of the Lord which we are, awaken our faith in order to enter into the presence of him who awaits us.
This sentence refers to the first part of the Mass, when the priest "gathers" together all the prayers of the faithful and prepares the altar for the sacrifice of the Mass.
The Catechism then continues with the comparison, relating how contemplative prayer is further connected to the Mas:.
We let our masks fall and turn our hearts back to the Lord who loves us, so as to hand ourselves over to him as an offering to be purified and transformed.
This refers to the second part of the Mass, when the priest offers the Eucharistic sacrifice, offering it to God.
In this way, contemplative prayer is an "offering," where we offer ourselves to God to be "purified and transformed."
Contemplative prayer is truly "Eucharistic," a method of prayer that mirrors the celebration of the Mass.