Finns have some Y-DNA, haplogroup N in them I believe. Haplogroup N is a Uralic Mongoloid patrilineal signature.
Also, Haplogroup N is a brother patriline of Haplogroup O (East Asians and SEAs). Meaning at one point in human history, they were basically the same tribe (Haplogroup NO) before they split off in different direction.
So yes, in that sense, Finns are a small part Asian. Much like Russians, some Mongolian (Y-DNA haplogroup C3) genetic influence as well as culture.
"Haplogroup N (M231) is a
Y-chromosome DNA haplogroup defined by the presence of the
single-nucleotide polymorphism (SNP) marker M231.
[Phylogenetics 1]
It is most commonly found in males originating from northern
Eurasia. It also has been observed at lower frequencies in populations native to other regions, including the
Balkans,
Central Asia,
East Asia, and
Southeast Asia."
Haplogroup N has a wide geographic distribution throughout northern Eurasia, and it also has been observed occasionally in other areas, including Central Asia and the Balkans.
It has been found with greatest frequency among
indigenous peoples of Russia, including
Finnic peoples,
Mari,
Udmurt,
Komi,
Khanty,
Mansi,
Nenets,
Nganasans,
Turkic peoples (Yakuts, Dolgans, Khakasses, Tuvans, Tatars, Chuvashes,
etc.),
Buryats,
Tungusic peoples (
Evenks,
Evens,
Negidals,
Nanais,
etc.),
Yukaghirs,
Luoravetlans (Chukchis, Koryaks), and
Siberian Eskimos, but certain subclades are very common in
Finland,
Estonia,
Latvia, and
Lithuania, and other subclades are found at low frequency in
China (Yi, Naxi, Lhoba,
Han Chinese,
etc.)