This will use the widest verse for deciding the width and center the poem.
\documentclass{article}
\usepackage{verse,varwidth,environ}
\usepackage{lipsum} % just for the example
\newsavebox{\versebox}
\NewEnviron{cverse}{%
\setlength{\leftmargini}{0pt}%
\begin{lrbox}{\versebox}
\begin{varwidth}{\textwidth}
\begin{verse}
\BODY
\end{verse}
\end{varwidth}
\end{lrbox}%
\begin{verse}[\wd\versebox]
\BODY
\end{verse}
}
\begin{document}
\lipsum[2]
\begin{cverse}
There was an old party of Lyme \\
Who married three wives at one time. \\
\vin When asked: ‘Why the third?’ \\
\vin He replied: ‘One’s absurd, \\
And bigamy, sir, is a crime.’
\end{cverse}
\begin{cverse}
Clof, clop, cloch,\\
cloffete,\\
cloppete,\\
clocchette,\\
chchch\dots\\
\`E gi\`u,\\
nel cortile,\\
la povera\\
fontana\\
malata;\\
che spasimo!\\
sentirla\\
tossire.\\
Tossisce,\\
tossisce,\\
un poco\\
si tace\dots\\
di nuovo.\\
tossisce.
\end{cverse}
\end{document}
The second poem is the start of Aldo Palazzeschi's “La fontana malata” (The sick fountain).

I used varwidth
for measuring the width; because of how verse
works, we need to set \leftmargini
to zero (the setting is inside cverse
, so it's local).
If you need footnotes in poems, change the code above into
\newsavebox{\versebox}
\newcounter{mockfootnote}
\NewEnviron{cverse}{%
\setlength{\leftmargini}{0pt}%
\begin{lrbox}{\versebox}
% remember the current value of footnote
\setcounter{mockfootnote}{\value{footnote}}%
% disable footnotes in the first pass over the poem
\renewcommand\footnote[1]{\stepcounter{mockfootnote}\textsuperscript{\themockfootnote}}%
\begin{varwidth}{\textwidth}
\begin{verse}
\BODY
\end{verse}
\end{varwidth}
\end{lrbox}%
\begin{verse}[\wd\versebox]
\BODY
\end{verse}
}
\versewidth
?