1833 newspaper report, unfortunately very little is known.

http://www.historyofuptonbychester.org.uk/canalmyst.html

There is a map, I think I've seen it on WikiWirral of the waterfront of Birkenhead showing a line more or less along the river, if my memory serves me says "proposed canal".

When a canal was proposed from Chester to the Mersey (at what became Ellesmere Port) the plans were shown on a map produced by J. Hunter. Cheshire Record Office dates this map as 1770. The canal route is shown with the words “Proposed Canal 8 Miles ½”.

When it was later proposed to build a canal from Ellesmere Port to Woodside, Birkenhead, the original Hunter map was re-used with hardly any updating. The now-existing Chester to Ellesmere Port canal (the earlier proposal) was shown as the “New Canal 9 Ms”. The proposed route from Ellesmere Port to Birkenhead was labelled “Proposed Extention of Canal 9 Miles”. The date on my copy of this map is difficult to read. I think it shows 1820 but there is a possibility that it shows 1840. Neither date tallies with that newspaper article. There may have been several proposals over the years and this map may have accompanied another one.

My copy is the same map (differently cropped) and has 1820 added by hand, which although it doesn't confirm it as 1820, would confirm your suspicions...

I can't say when the map was made, however I can't find any mention of a proposed canal, Birkenhead/ Wallasey Pool/ Chester, prior to 1833 in the newspapers. The 1833 canal is mentioned throughout various papers around the country.
Below a couple of snippets, October/November 1833 Liverpool Mercury.

Possibly tied in to my query
Maybe it was another scheme to get more money from Liverpool after the previous success yollers post

It's some while since I last looked at the Hunter maps used for the two proposed canals - 1770 for Chester to Ellesmere Port and 1820 for Ellesmere Port to Woodside.
Looking again at them I conclude that neither was produced specifically for the proposed canals. The planned canal routes were added to existing maps.
The 1770 map is dedicated ("most respectfully inscribed") to the "Proprietors and Captains of the Dublin & Parkgate Packets". It shows six vessels in the River Dee and one in the Mersey.
The 1820 map is dedicated to the "Proprietors and Captains of the Dublin & Liverpool Packets" and still shows the same six vessels in the Dee but now has eight vessels in the Mersey. (Parkgate was finished as a Dublin passenger port by 1815.)
Land features show no changes at all between 1770 and 1820.
I am reasonably confident that the 1820 Hunter map was the one used to accompany the 1833 plan for a canal from Ellesmere Port to Woodside.