Adenosine is synthesised from Inosine Monophosphate (IMP) as the nucleotide form (adenosine monophosphate).
IMP does not accumulate in the cell but is rapidly converted to AMP and GMP. AMP, which differs from IMP only in the replacement of its 6-keto group by an amino is synthesized in a two-reaction pathway.
In 1948, John Buchanan obtained the first clues as to how nucleic
acids and their components are synthesized by describing the synthesis
of purine ribonucleotides.
Buchanan investigated how this process occurs de novo by feeding a variety of isotopically labelled compounds to pigeons and chemically determining the positions of the labelled atoms in their excreted uric acid (a purine).

The biosynthetic origins of purine ring atoms. Note that C4, C5, and N7 come from a single glycine molecule but each of the other atoms is derived from an independent precursor.
The actual pathway by which these precursors are incorporated into the purine ring, was elucidated in subsequent investigations performed largely by Buchanan and by G. Robert Greenberg. These investigations showed that the initially synthesized purine derivative is inosine monophosphate (IMP),
(Here I am not going to explain the biosynthesis of IMP but focus on the former)
IMP is converted to AMP (or GMP) in separate two-reaction pathways:
In the first reaction, aspartate’s amino group is linked to IMP in a reaction driven by the hydrolysis of GTP to GDP + Pi to yield adenylosuccinate.
In the second reaction, adenylosuccinate lyase eliminates fumarate from adenylosuccinate to form AMP (adenosine monophosphate).

It is important to note that free adenine (and guanine) can be reconverted to corresponding ribonucleotides through salvage pathways requiring the enzyme Adenine phosphoribosyltransferase (APRT) (mediates AMP formation)
References
Voet and Voet Biochemistry: Section 28-1. Synthesis of Purine Ribonucleotides